Hot electrons in superlattices: quantum transport versus Boltzmann equation
DEFF Research Database (Denmark)
Wacker, Andreas; Jauho, Antti-Pekka; Rott, S.;
1999-01-01
A self-consistent solution of the transport equation is presented for semiconductor superlattices within different approaches: (i) a full quantum transport model based on nonequilibrium Green functions, (ii) the semiclassical Boltzmann equation for electrons in a miniband, and (iii) Boltzmann...
A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation
Energy Technology Data Exchange (ETDEWEB)
Larsen, Edward
2013-06-17
The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.
Tervo, J; Frank, M; Herty, M
2016-01-01
The paper considers a coupled system of linear Boltzmann transport equation (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in radiation therapy. The evolution of charged particles (e.g. electrons and positrons) are in practice often modelled using the CSDA version of BTE because of the so-called forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems for numerical methods. First, we prove the existence and uniqueness of solutions, under sufficient criteria and in appropriate $L^2$-based spaces, of a single (particle) CSDA-equation by using two complementary techniques, the Lions-Lax-Milgram Theorem (variational approach), and the theory evolution operators (semigroup approach). The necessary a priori estimates are shown. In addition, we prove the corresponding results and estimates for the system of coupled transport equat...
Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study
Kaiser, J.; Feng, T.; Maassen, J.; Wang, X.; Ruan, X.; Lundstrom, M.
2017-01-01
Steady-state thermal transport in nanostructures with dimensions comparable to the phonon mean-free-path is examined. Both the case of contacts at different temperatures with no internal heat generation and contacts at the same temperature with internal heat generation are considered. Fourier's law results are compared to finite volume method solutions of the phonon Boltzmann equation in the gray approximation. When the boundary conditions are properly specified, results obtained using Fourier's law without modifying the bulk thermal conductivity are in essentially exact quantitative agreement with the phonon Boltzmann equation in the ballistic and diffusive limits. The errors between these two limits are examined in this paper. For the four cases examined, the error in the apparent thermal conductivity as deduced from a correct application of Fourier's law is less than 6%. We also find that the Fourier's law results presented here are nearly identical to those obtained from a widely used ballistic-diffusive approach but analytically much simpler. Although limited to steady-state conditions with spatial variations in one dimension and to a gray model of phonon transport, the results show that Fourier's law can be used for linear transport from the diffusive to the ballistic limit. The results also contribute to an understanding of how heat transport at the nanoscale can be understood in terms of the conceptual framework that has been established for electron transport at the nanoscale.
Cobos, Agustín C.; Poma, Ana L.; Alvarez, Guillermo D.; Sanz, Darío E.
2016-10-01
We introduce an alternative method to calculate the steady state solution of the angular photon flux after a numerical evolution of the time-dependent Boltzmann transport equation (BTE). After a proper discretization the transport equation was converted into an ordinary system of differential equations that can be iterated as a weighted Richardson algorithm. As a different approach, in this work the time variable regulates the iteration process and convergence criteria is based on physical parameters. Positivity and convergence was assessed from first principles and a modified Courant-Friedrichs-Lewy condition was devised to guarantee convergence. The Penelope Monte Carlo method was used to test the convergence and accuracy of our approach for different phase space discretizations. Benchmarking was performed by calculation of total fluence and photon spectra in different one-dimensional geometries irradiated with 60Co and 6 MV photon beams and radiological applications were devised.
An introduction to the Boltzmann equation and transport processes in gases
Energy Technology Data Exchange (ETDEWEB)
Kremer, Gilberto Medeiros [Universidade Federal do Parana, Curitiba (Brazil). Dept. de Fisica
2010-07-01
This book deals with the classical kinetic theory of gases. Its aim is to present the basic principles of this theory within an elementary framework and from a more rigorous approach based on the Boltzmann equation. The subjects are presented in a self-contained manner such that the readers can understand and learn some methods used in the kinetic theory of gases in order to investigate the Boltzmann equation. It is expected that this book could be useful as a textbook for students and researchers who are interested in the principles of the Boltzmann equation and in the methods used in the kinetic theory of gases. (orig.)
An introduction to the theory of the Boltzmann equation
Harris, Stewart
2011-01-01
Boltzmann's equation (or Boltzmann-like equations) appears extensively in such disparate fields as laser scattering, solid-state physics, nuclear transport, and beyond the conventional boundaries of physics and engineering, in the fields of cellular proliferation and automobile traffic flow. This introductory graduate-level course for students of physics and engineering offers detailed presentations of the basic modern theory of Boltzmann's equation, including representative applications using both Boltzmann's equation and the model Boltzmann equations developed within the text. It emphasizes
Energy Technology Data Exchange (ETDEWEB)
Hagelaar, G J M; Pitchford, L C [Centre de Physique des Plasmas et de leurs Applications de Toulouse, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 (France)
2005-11-15
Fluid models of gas discharges require the input of transport coefficients and rate coefficients that depend on the electron energy distribution function. Such coefficients are usually calculated from collision cross-section data by solving the electron Boltzmann equation (BE). In this paper we present a new user-friendly BE solver developed especially for this purpose, freely available under the name BOLSIG+, which is more general and easier to use than most other BE solvers available. The solver provides steady-state solutions of the BE for electrons in a uniform electric field, using the classical two-term expansion, and is able to account for different growth models, quasi-stationary and oscillating fields, electron-neutral collisions and electron-electron collisions. We show that for the approximations we use, the BE takes the form of a convection-diffusion continuity-equation with a non-local source term in energy space. To solve this equation we use an exponential scheme commonly used for convection-diffusion problems. The calculated electron transport coefficients and rate coefficients are defined so as to ensure maximum consistency with the fluid equations. We discuss how these coefficients are best used in fluid models and illustrate the influence of some essential parameters and approximations.
Energy Technology Data Exchange (ETDEWEB)
Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: roj@correo.azc.uam.mx
2003-07-01
There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S{sub 4} with expansions of the dispersion cross sections until P{sub 3} order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)
Energy Technology Data Exchange (ETDEWEB)
Merton, S.R. [Computational Physics Group, AWE, Aldermaston, Reading, Berkshire RG7 4PR (United Kingdom)], E-mail: simon.merton@awe.co.uk; Pain, C.C. [Computational Physics and Geophysics Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2A7 (United Kingdom); Smedley-Stevenson, R.P. [Computational Physics Group, AWE, Aldermaston, Reading, Berkshire RG7 4PR (United Kingdom); Buchan, A.G.; Eaton, M.D. [Computational Physics and Geophysics Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2A7 (United Kingdom)
2008-09-15
This paper describes the development of two optimal discontinuous finite element (FE) Riemann methods and their application to the one-speed Boltzmann transport equation in the steady-state. The proposed methods optimise the amount of dissipation applied in the streamline direction. This dissipation is applied within an element using a novel Riemann FE method, which is based on an analogy between control volume discretisation methods and finite element methods when integration by parts is applied to the transport terms. In one-dimension the optimal finite element solutions match the analytical solution exactly at each outlet node. Both schemes couple elements in space via a Riemann approach. The first of the two schemes is a Petrov-Galerkin (PG) method which introduces dissipation via the equation residual. The second scheme uses a streamline diffusion stabilisation term in the discretisation. These two methods provide a discontinuous Petrov-Galerkin (DPG) scheme that can stabilise an element across the full range of radiation regimes, obtaining robust solutions with suppressed oscillation. Three basis functions in angle of particle travel have been implemented in an optimal DPG Riemann solver, which include the P{sub N} (spherical harmonic), S{sub N} (discrete ordinate) and LW{sub N} (linear octahedral wavelet) angular expansions. These methods are applied to a series of demanding two-dimensional radiation transport problems.
Dujko, S.; Ebert, U.; White, R.D.; Petrović, Z.L.
2010-01-01
A comprehensive investigation of electron transport in N$_{2}$-O$_{2}$ mixtures has been carried out using a multi term theory for solving the Boltzmann equation and Monte Carlo simulation technique instead of conventional two-term theory often employed in plasma modeling community. We focus on the
Romano, Giuseppe; Esfarjani, Keivan; Strubbe, David A.; Broido, David; Kolpak, Alexie M.
2016-01-01
Nanostructured materials exhibit low thermal conductivity because of the additional scattering due to phonon-boundary interactions. As these interactions are highly sensitive to the mean free path (MFP) of phonons, MFP distributions in nanostructures can be dramatically distorted relative to bulk. Here we calculate the MFP distribution in periodic nanoporous Si for different temperatures, using the recently developed MFP-dependent Boltzmann transport equation. After analyzing the relative contribution of each phonon branch to thermal transport in nanoporous Si, we find that at room temperature optical phonons contribute 17 % to heat transport, compared to 5 % in bulk Si. Interestingly, we observe a constant thermal conductivity over the range 200 K acoustic phonons with long intrinsic MFP and the temperature dependence of the heat capacity. Our findings, which are in qualitative agreement with the temperature trend of thermal conductivities measured in nanoporous Si-based systems, shed light on the origin of the reduction of thermal conductivity in nanostructured materials and demonstrate the necessity of multiscale heat transport engineering, in which the bulk material and geometry are optimized concurrently.
Energy Technology Data Exchange (ETDEWEB)
Buchan, Andrew G., E-mail: andrew.buchan@imperial.ac.uk [Applied Modelling and Computational Group, Department of Earth Science and Engineering, Imperial College of Science, Technology and Medicine (United Kingdom); Merton, Simon R. [AWE, Aldermaston, Reading RG7 4PR (United Kingdom); Pain, Christopher C. [Applied Modelling and Computational Group, Department of Earth Science and Engineering, Imperial College of Science, Technology and Medicine (United Kingdom); Smedley-Stevenson, Richard P. [AWE, Aldermaston, Reading RG7 4PR (United Kingdom)
2011-05-15
In this paper a method for resolving the various boundary conditions (BCs) for the first order Boltzmann transport equation (BTE) is described. The approach has been formulated to resolve general BCs using an arbitrary angular approximation method within any weighted residual finite element formulation. The method is based on a Riemann decomposition which is used to decompose the particles' angular dependence into in-coming and out-going information through a surface. This operation recasts the flux into a Riemann space which is used directly to remove any incoming information, and thus satisfy void boundary conditions. The method is then extended by its coupling with a set of mapping operators that redirect the outgoing flux to form incoming images resembling other specified boundary conditions. These operators are based on Galerkin projections and are defined to enable reflective and diffusive (white) BCs to be resolved. A small number of numerical examples are then presented to demonstrate the method's ability in resolving void, reflective and white BCs. These examples have been chosen in order to show the method working for arbitrary angled surfaces. Furthermore, as the method has been designed for an arbitrary angular approximation, both S{sub N} and P{sub N} calculations are presented.
Fiorentini, Mattia; Bonini, Nicola
2016-08-01
We present a first-principles computational approach to calculate thermoelectric transport coefficients via the exact solution of the linearized Boltzmann transport equation, also including the effect of nonequilibrium phonon populations induced by a temperature gradient. We use density functional theory and density functional perturbation theory for an accurate description of the electronic and vibrational properties of a system, including electron-phonon interactions; carriers' scattering rates are computed using standard perturbation theory. We exploit Wannier interpolation (both for electronic bands and electron-phonon matrix elements) for an efficient sampling of the Brillouin zone, and the solution of the Boltzmann equation is achieved via a fast and stable conjugate gradient scheme. We discuss the application of this approach to n -doped silicon. In particular, we discuss a number of thermoelectric properties such as the thermal and electrical conductivities of electrons, the Lorenz number and the Seebeck coefficient, including the phonon drag effect, in a range of temperatures and carrier concentrations. This approach gives results in good agreement with experimental data and provides a detailed characterization of the nature and the relative importance of the individual scattering mechanisms. Moreover, the access to the exact solution of the Boltzmann equation for a realistic system provides a direct way to assess the accuracy of different flavors of relaxation time approximation, as well as of models that are popular in the thermoelectric community to estimate transport coefficients.
Kinetic Boltzmann, Vlasov and Related Equations
Sinitsyn, Alexander; Vedenyapin, Victor
2011-01-01
Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in
The Non-Classical Boltzmann Equation, and Diffusion-Based Approximations to the Boltzmann Equation
Frank, Martin; Larsen, Edward W; Vasques, Richard
2014-01-01
We show that several diffusion-based approximations (classical diffusion or SP1, SP2, SP3) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors - no truncation errors.
Energy Technology Data Exchange (ETDEWEB)
Lloyd, S. A. M.; Ansbacher, W. [Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada); Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada) and Department of Medical Physics, British Columbia Cancer Agency-Vancouver Island Centre, Victoria, British Columbia V8R 6V5 (Canada)
2013-01-15
Purpose: Acuros external beam (Acuros XB) is a novel dose calculation algorithm implemented through the ECLIPSE treatment planning system. The algorithm finds a deterministic solution to the linear Boltzmann transport equation, the same equation commonly solved stochastically by Monte Carlo methods. This work is an evaluation of Acuros XB, by comparison with Monte Carlo, for dose calculation applications involving high-density materials. Existing non-Monte Carlo clinical dose calculation algorithms, such as the analytic anisotropic algorithm (AAA), do not accurately model dose perturbations due to increased electron scatter within high-density volumes. Methods: Acuros XB, AAA, and EGSnrc based Monte Carlo are used to calculate dose distributions from 18 MV and 6 MV photon beams delivered to a cubic water phantom containing a rectangular high density (4.0-8.0 g/cm{sup 3}) volume at its center. The algorithms are also used to recalculate a clinical prostate treatment plan involving a unilateral hip prosthesis, originally evaluated using AAA. These results are compared graphically and numerically using gamma-index analysis. Radio-chromic film measurements are presented to augment Monte Carlo and Acuros XB dose perturbation data. Results: Using a 2% and 1 mm gamma-analysis, between 91.3% and 96.8% of Acuros XB dose voxels containing greater than 50% the normalized dose were in agreement with Monte Carlo data for virtual phantoms involving 18 MV and 6 MV photons, stainless steel and titanium alloy implants and for on-axis and oblique field delivery. A similar gamma-analysis of AAA against Monte Carlo data showed between 80.8% and 87.3% agreement. Comparing Acuros XB and AAA evaluations of a clinical prostate patient plan involving a unilateral hip prosthesis, Acuros XB showed good overall agreement with Monte Carlo while AAA underestimated dose on the upstream medial surface of the prosthesis due to electron scatter from the high-density material. Film measurements
Di, Shaoyan; Shen, Lei; Chang, Pengying; Zhao, Kai; Lu, Tiao; Du, Gang; Liu, Xiaoyan
2017-04-01
A deterministic time-dependent Boltzmann transport equation (BTE) solver is employed to carry out a comparison work among 10 nm double-gate n-type MOSFETs with channel materials of Si, In0.53Ga0.47As, and GaSb in different surface orientations. Results show that the GaSb device has the highest drive current, while scattering affects carrier transport in the Si device the most. The InGaAs device exhibits the highest injection velocity but suffers from the density of state (DOS) bottleneck seriously.
Energy Technology Data Exchange (ETDEWEB)
Mirza, Anwar M. [Department of Computer Science, National University of Computer and Emerging Sciences, NUCES-FAST, A.K. Brohi Road, H-11, Islamabad (Pakistan)], E-mail: anwar.m.mirza@gmail.com; Iqbal, Shaukat [Faculty of Computer Science and Engineering, Ghulam Ishaq Khan (GIK) Institute of Engineering Science and Technology, Topi-23460, Swabi (Pakistan)], E-mail: shaukat@giki.edu.pk; Rahman, Faizur [Department of Physics, Allama Iqbal Open University, H-8 Islamabad (Pakistan)
2007-07-15
A spatially adaptive grid-refinement approach has been investigated to solve the even-parity Boltzmann transport equation. A residual based a posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local particle balance has been considered as an error assessment criterion. To implement the adaptive approach, a computer program ADAFENT (adaptive finite elements for neutron transport) has been developed to solve the second order even-parity Boltzmann transport equation using K{sup +} variational principle for slab geometry. The program has a core K{sup +} module which employs Lagrange polynomials as spatial basis functions for the finite element formulation and Legendre polynomials for the directional dependence of the solution. The core module is called in by the adaptive grid generator to determine local gradients and residuals to explore the possibility of grid refinements in appropriate regions of the problem. The a posteriori error estimation scheme has been implemented in the outer grid refining iteration module. Numerical experiments indicate that local errors are large in regions where the flux gradients are large. A comparison of the spatially adaptive grid-refinement approach with that of uniform meshing approach for various benchmark cases confirms its superiority in greatly enhancing the accuracy of the solution without increasing the number of unknown coefficients. A reduction in the local errors of the order of 10{sup 2} has been achieved using the new approach in some cases.
Lattice Boltzmann solver of Rossler equation
Institute of Scientific and Technical Information of China (English)
GuangwuYAN; LiRUAN
2000-01-01
We proposed a lattice Boltzmann model for the Rossler equation. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler equation on the mesescopic scale. The numerical results show the method can be used to simulate Rossler equation.
Quantum corrections for Boltzmann equation
Institute of Scientific and Technical Information of China (English)
M.; Levy; PETER
2008-01-01
We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.
Coarse-grained transport of a turbulent flow via moments of the Reynolds-averaged Boltzmann equation
Abramov, Rafail V
2015-01-01
Here we introduce new coarse-grained variables for a turbulent flow in the form of moments of its Reynolds-averaged Boltzmann equation. With the exception of the collision moments, the transport equations for the new variables are identical to the usual moment equations, and thus naturally lend themselves to the variety of already existing closure methods. Under the anelastic turbulence approximation, we derive equations for the Reynolds-averaged turbulent fluctuations around the coarse-grained state. We show that the global relative entropy of the coarse-grained state is bounded from above by the Reynolds average of the fine-grained global relative entropy, and thus obeys the time decay bound of Desvillettes and Villani. This is similar to what is observed in the rarefied gas dynamics, which makes the Grad moment closure a good candidate for truncating the hierarchy of the coarse-grained moment equations. We also show that, under additional assumptions on the form of the coarse-grained collision terms, one a...
Boltzmann equations for neutrinos with flavor mixings
Yamada, Shoichi
2000-01-01
With a view of applications to the simulations of supernova explosion and proto neutron star cooling, we derive the Boltzmann equations for the neutrino transport with the flavor mixing based on the real time formalism of the nonequilibrium field theory and the gradient expansion of the Green function. The relativistic kinematics is properly taken into account. The advection terms are derived in the mean field approximation for the neutrino self-energy whiles the collision terms are obtained ...
Energy Technology Data Exchange (ETDEWEB)
Bankovic, A., E-mail: ana.bankovic@gmail.com [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia); Dujko, S. [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia); Centrum Wiskunde and Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam (Netherlands); ARC Centre for Antimatter-Matter Studies, School of Engineering and Physical Sciences, James Cook University, Townsville, QLD 4810 (Australia); White, R.D. [ARC Centre for Antimatter-Matter Studies, School of Engineering and Physical Sciences, James Cook University, Townsville, QLD 4810 (Australia); Buckman, S.J. [ARC Centre for Antimatter-Matter Studies, Australian National University, Canberra, ACT 0200 (Australia); Petrovic, Z.Lj. [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia)
2012-05-15
This work reports on a new series of calculations of positron transport properties in molecular hydrogen under the influence of spatially homogeneous electric field. Calculations are performed using a Monte Carlo simulation technique and multi term theory for solving the Boltzmann equation. Values and general trends of the mean energy, drift velocity and diffusion coefficients as a function of the reduced electric field E/n{sub 0} are reported here. Emphasis is placed on the explicit and implicit effects of positronium (Ps) formation on the drift velocity and diffusion coefficients. Two important phenomena arise; first, for certain regions of E/n{sub 0} the bulk and flux components of the drift velocity and longitudinal diffusion coefficient are markedly different, both qualitatively and quantitatively. Second, and contrary to previous experience in electron swarm physics, there is negative differential conductivity (NDC) effect in the bulk drift velocity component with no indication of any NDC for the flux component. In order to understand this atypical manifestation of the drift and diffusion of positrons in H{sub 2} under the influence of electric field, the spatially dependent positron transport properties such as number of positrons, average energy and velocity and spatially resolved rate for Ps formation are calculated using a Monte Carlo simulation technique. The spatial variation of the positron average energy and extreme skewing of the spatial profile of positron swarm are shown to play a central role in understanding the phenomena.
Lattice Boltzmann equation for relativistic quantum mechanics.
Succi, Sauro
2002-03-15
Relativistic versions of the quantum lattice Boltzmann equation are discussed. It is shown that the inclusion of nonlinear interactions requires the standard collision operator to be replaced by a pair of dynamic fields coupling to the relativistic wave function in a way which can be described by a multicomponent complex lattice Boltzmann equation.
Inelastic Quantum Transport in Superlattices: Success and Failure of the Boltzmann Equation
DEFF Research Database (Denmark)
Wacker, Andreas; Jauho, Antti-Pekka; Rott, Stephan;
1999-01-01
Electrical transport in semiconductor superlattices is studied within a fully self-consistent quantum transport model based on nonequilibrium Green functions, including phonon and impurity scattering. We compute both the drift-velocity-held relation and the momentum distribution function covering...
Priimak, Dmitri
2014-01-01
We present finite differences numerical algorithm for solving 2D spatially homogeneous Boltzmann transport equation for semiconductor superlattices (SL) subject to time dependant electric field along SL axis and constant perpendicular magnetic field. Algorithm is implemented in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPUs. We compare performance and merits of one implementation versus another and discuss various methods of optimization.
New approach to the solution of the Boltzmann radiation transport equation
Boffi, Vinicio C.; Dunn, William L.
1987-03-01
Transport monodimensional stationary solutions for the angular space-energy neutron flux, of interest in radiation penetration problems, are studied by Green's function method. Explicit analytical results for the spatial moments of the sought solution are obtained for the case of an isotropically scattering slab of infinite thickness and of a continuous slowing down model in energy.
Classical Boltzmann equation and high-temperature QED
Brandt, F. T.; Ferreira, R. B.; Thuorst, J. F.
2015-01-01
The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of quantum electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the ...
The Boltzmann equation in the difference formulation
Energy Technology Data Exchange (ETDEWEB)
Szoke, Abraham [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks III, Eugene D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2015-05-06
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
THREE WAY DECOMPOSITION FOR THE BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
Ilgis Ibragimov; Sergej Rjasanow
2009-01-01
The initial value problem for the spatially homogeneous Boltzmann equation is considered. A deterministic numerical scheme for this problem is developed by the use of the three way decomposition of the unknown function as well as of the collision integral. On this way, almost linear complexity of the algorithm is achieved. Some numerical examples are presented.
Entropic lattice Boltzmann model for Burgers's equation.
Boghosian, Bruce M; Love, Peter; Yepez, Jeffrey
2004-08-15
Entropic lattice Boltzmann models are discrete-velocity models of hydrodynamics that possess a Lyapunov function. This feature makes them useful as nonlinearly stable numerical methods for integrating hydrodynamic equations. Over the last few years, such models have been successfully developed for the Navier-Stokes equations in two and three dimensions, and have been proposed as a new category of subgrid model of turbulence. In the present work we develop an entropic lattice Boltzmann model for Burgers's equation in one spatial dimension. In addition to its pedagogical value as a simple example of such a model, our result is actually a very effective way to simulate Burgers's equation in one dimension. At moderate to high values of viscosity, we confirm that it exhibits no trace of instability. At very small values of viscosity, however, we report the existence of oscillations of bounded amplitude in the vicinity of the shock, where gradient scale lengths become comparable with the grid size. As the viscosity decreases, the amplitude at which these oscillations saturate tends to increase. This indicates that, in spite of their nonlinear stability, entropic lattice Boltzmann models may become inaccurate when the ratio of gradient scale length to grid spacing becomes too small. Similar inaccuracies may limit the utility of the entropic lattice Boltzmann paradigm as a subgrid model of Navier-Stokes turbulence.
General relativistic Boltzmann equation, I: Covariant treatment
Debbasch, F.; van Leeuwen, W.A.
2009-01-01
This series of two articles aims at dissipating the rather dense haze existing in the present literature around the General Relativistic Boltzmann equation. In this first article, the general relativistic one-particle distribution function in phase space is defined as an average of delta functions.
Full Boltzmann equations for leptogenesis including scattering
Hahn-Woernle, F; Wong, Y Y Y
2009-01-01
We study the evolution of a cosmological baryon asymmetry produced via leptogenesis by means of the full classical Boltzmann equations, without the assumption of kinetic equilibrium and including all quantum statistical factors. Beginning with the full mode equations we derive the usual equations of motion for the right-handed neutrino number density and integrated lepton asymmetry, and show explicitly the impact of each assumption on these quantities. For the first time, we investigate also the effects of scattering of the right-handed neutrino with the top quark to leading order in the Yukawa couplings by means of the full Boltzmann equations. We find that in our full Boltzmann treatment the final lepton asymmetry can be suppressed by as much as a factor of 1.5 in the weak wash-out regime (K1), the full Boltzmann treatment and the integrated approach give nearly identical final lepton asymmetries (within 10 % of each other at K>3). Finally, we show that the opposing effects of quantum statistics on decays/i...
Classical Boltzmann equation and high-temperature QED
Brandt, F. T.; Ferreira, R. B.; Thuorst, J. F.
2015-02-01
The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of quantum electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the physics described by the leading high temperature limit of quantum electrodynamics can be obtained from the Boltzman transport equation. We also present some explicit examples of this interesting equivalence.
Classical Boltzmann equation and high-temperature QED
Brandt, F T; Thuorst, J F
2015-01-01
The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of Quantum Electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the physics described by the leading high temperature limit of Quantum Electrodynamics can be obtained from the Boltzman transport equation. We also present some explicit examples of this interesting equivalence.
Boltzmann transport calculation of collinear spin transport on short timescales
Nenno, Dennis M.; Kaltenborn, Steffen; Schneider, Hans Christian
2016-09-01
A spin-dependent Boltzmann transport equation is used to describe charge and spin dynamics resulting from the excitation of hot electrons in a ferromagnet/normal metal heterostructure. As the microscopic Boltzmann equation works with k -dependent distribution functions, it can describe far-from-equilibrium excitations, which are outside the scope of drift-diffusion theories. We study different scenarios for spin-dependent carrier injection into a nonmagnetic metal using an effectively two-dimensional phase space. While the charge signal is robust for various excitation schemes, the shape of the resulting spin current/density depends critically on the interplay between transport and scattering, and on the energetic distribution of the injected carriers. Our results imply that the energy dependence of the injected hot electrons has a decisive effect on the spin dynamics.
On the full Boltzmann equations for Leptogenesis
Garayoa, J; Pinto, T; Rius, N; Vives, O
2009-01-01
We consider the full Boltzmann equations for standard and soft leptogenesis, instead of the usual integrated Boltzmann equations which assume kinetic equilibrium for all species. Decays and inverse decays may be inefficient for thermalising the heavy-(s)neutrino distribution function, leading to significant deviations from kinetic equilibrium. We analyse the impact of using the full kinetic equations in the case of a previously generated lepton asymmetry, and find that the washout of this initial asymmetry due to the interactions of the right-handed neutrino is larger than when calculated via the integrated equations. We also solve the full Boltzmann equations for soft leptogenesis, where the lepton asymmetry induced by the soft SUSY-breaking terms in sneutrino decays is a purely thermal effect, since at T=0 the asymmetry in leptons cancels the one in sleptons. In this case, we obtain that in the weak washout regime (K ~< 1) the final lepton asymmetry can change up to a factor four with respect to previous...
On the full Boltzmann equations for leptogenesis
Energy Technology Data Exchange (ETDEWEB)
Garayoa, J.; Pastor, S.; Pinto, T.; Rius, N.; Vives, O., E-mail: garayoa@ific.uv.es, E-mail: pastor@ific.uv.es, E-mail: teguayco@gmail.com, E-mail: nuria@ific.uv.es, E-mail: vives@ific.uv.es [Depto. de Física Teórica and IFIC, Universidad de Valencia-CSIC, Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia (Spain)
2009-09-01
We consider the full Boltzmann equations for standard and soft leptogenesis, instead of the usual integrated Boltzmann equations which assume kinetic equilibrium for all species. Decays and inverse decays may be inefficient for thermalising the heavy-(s)neutrino distribution function, leading to significant deviations from kinetic equilibrium. We analyse the impact of using the full kinetic equations in the case of a previously generated lepton asymmetry, and find that the washout of this initial asymmetry due to the interactions of the right-handed neutrino is larger than when calculated via the integrated equations. We also solve the full Boltzmann equations for soft leptogenesis, where the lepton asymmetry induced by the soft SUSY-breaking terms in sneutrino decays is a purely thermal effect, since at T = 0 the asymmetry in leptons cancels the one in sleptons. In this case, we obtain that in the weak washout regime (K ∼< 1) the final lepton asymmetry can change up to a factor four with respect to previous estimates.
Global Solutions of the Boltzmann Equation Over {{R}^D} Near Global Maxwellians with Small Mass
Bardos, Claude; Gamba, Irene M.; Golse, François; Levermore, C. David
2016-09-01
We study the dynamics defined by the Boltzmann equation set in the Euclidean space {{R}^D} in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.
Efficient Asymptotic Preserving Deterministic methods for the Boltzmann Equation
2011-04-01
important role in modelling granular gases (Bobylev et al., 2000), charged particles in semiconductors (Markowich et al., 1989), neutron transport (Jin...1.6 The splitting approach 1 THE BOLTZMANN EQUATION • Granular gas models : particles undergo inelastic collisions . Energy is dissipated by the model ...A model for collision processes in gases i. small amplitute processes in charged and neutral one component systems. Phys. Rev., 94:511–525. 8
A Fluctuating Lattice Boltzmann Method for the Diffusion Equation
Wagner, Alexander J
2016-01-01
We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.
Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics.
García de Soria, María Isabel; Maynar, Pablo; Schehr, Grégory; Barrat, Alain; Trizac, Emmanuel
2008-05-01
We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions.
Energy Technology Data Exchange (ETDEWEB)
Stoenescu, M.L.
1977-06-01
The terms in Boltzmann kinetic equation corresponding to elastic short range collisions, inelastic excitational collisions, coulomb interactions and electric field acceleration are evaluated numerically for a standard distribution function minimizing the computational volume by expressing the terms as linear combinations with recalculable coefficients, of the distribution function and its derivatives. The present forms are suitable for spatial distribution calculations.
Lattice Boltzmann method for the fractional advection-diffusion equation
Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Lattice Boltzmann method for the fractional advection-diffusion equation.
Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Soluble Boltzmann equations for internal state and Maxwell models
Futcher, E.; Hoare, M.R.; Hendriks, E.M.; Ernst, M.H.
1980-01-01
We consider a class of scalar nonlinear Boltzmann equations describing the evolution of a microcanonical ensemble in which sub-systems exchange internal energy ‘randomly’ in binary interactions. In the continuous variable version these models can equally be interpreted as Boltzmann equations for Ma
Comparison of Boltzmann Equations with Quantum Dynamics for Scalar Fields
Lindner, Manfred; Lindner, Manfred; Muller, Markus Michael
2006-01-01
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after relativistic heavy ion collisions. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this paper a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar Phi^4 quantum field theory in 3+1 space-time dimensions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Most notably, apart from quantitative discrepancies, on a qualitative level the universality observed for the Kadanoff-Baym equations is severely restricted in the case o...
Analysis of Jeans instability from Boltzmann equation
Kremer, Gilberto M
2015-01-01
The dynamics of self-gravitating fluids is analyzed within the framework of a collisionless Boltzmann equation in the presence of gravitational fields and Poisson equation. The equilibrium distribution function takes into account the expansion of the Universe and a pressureless fluid in the matter dominated Universe. Without invoking Jeans "swindle" a dispersion relation is obtained by considering small perturbations of the equilibrium values of the distribution function and gravitational potential. The collapse criterion -- which happens in an unstable region where the solution grows exponentially with time -- is determined from the dispersion relation. The collapse criterion in a static Universe occurs when the wavenumber $k$ is smaller than the Jeans wavenumber $k_J$, which was the solution found by Jeans. For an expanding Universe it is shown that this criterion is $k\\leq\\sqrt{7/6}\\,k_J$. As a consequence the ratio of the mass contained in a sphere of diameter equal to the wavelength $\\lambda=2\\pi/k$ to t...
Thermal equation of state for lattice Boltzmann gases
Institute of Scientific and Technical Information of China (English)
Ran Zheng
2009-01-01
The Galilean invaxiance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model axe proposed together with their rigorous theoretical background. From the viewpoint of group invariance,recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.
General relativistic Boltzmann equation, II: Manifestly covariant treatment
Debbasch, F.; van Leeuwen, W.A.
2009-01-01
In a preceding article we presented a general relativistic treatment of the derivation of the Boltzmann equation. The four-momenta occurring in this formalism were all on-shell four-momenta, verifying the mass-shell restriction p(2) = m(2)c(2). Due to this restriction, the resulting Boltzmann equati
Axisymmetric multiphase lattice Boltzmann method for generic equations of state
Reijers, S.A.; Gelderblom, H.; Toschi, F.
2016-01-01
We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid–gas density ratios up to 103. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation
Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation
Gamba, I. M.; Panferov, V.; Villani, C.
2007-01-01
For the spatially homogeneous Boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above, uniformly in time, by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.
Electric Conductivity from the solution of the Relativistic Boltzmann Equation
Puglisi, A; Greco, V
2014-01-01
We present numerical results of electric conductivity $\\sigma_{el}$ of a fluid obtained solving the Relativistic Transport Boltzmann equation in a box with periodic boundary conditions. We compute $\\sigma_{el}$ using two methods: the definition itself, i.e. applying an external electric field, and the evaluation of the Green-Kubo relation based on the time evolution of the current-current correlator. We find a very good agreement between the two methods. We also compare numerical results with analytic formulas in Relaxation Time Approximation (RTA) where the relaxation time for $\\sigma_{el}$ is determined by the transport cross section $\\sigma_{tr}$, i.e. the differential cross section weighted with the collisional momentum transfer. We investigate the electric conductivity dependence on the microscopic details of the 2-body scatterings: isotropic and anisotropic cross-section, and massless and massive particles. We find that the RTA underestimates considerably $\\sigma_{el}$; for example at screening masses $...
Energy Technology Data Exchange (ETDEWEB)
Uchaikin, V V; Sibatov, R T, E-mail: vuchaikin@gmail.com, E-mail: ren_sib@bk.ru [Ulyanovsk State University, 432000, 42 Leo Tolstoy str., Ulyanovsk (Russian Federation)
2011-04-08
The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start with the standard Boltzmann equation; averaging over photon frequencies leads to the appearance of a fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of a fractional material derivative in the equation is concordant with heavy tailed distribution of photon path lengths and with spatiotemporal coupling peculiar to the process. We discuss some methods of solving the obtained equation and demonstrate numerical results in some simple cases.
Analysis of spectral methods for the homogeneous Boltzmann equation
Filbet, Francis
2011-04-01
The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.
Celebrating Cercignani's conjecture for the Boltzmann equation
Villani, Cédric
2011-01-01
Cercignani\\'s conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann\\'s nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s. © American Institute of Mathematical Sciences.
Equations of motion of test particles for solving the spin-dependent Boltzmann-Vlasov equation
Xia, Yin; Li, Bao-An; Shen, Wen-Qing
2016-01-01
A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann-Vlasov equation is presented. Though the obtained EOMs are general, they are particularly useful in simulating nucleon spinor transport in heavy-ion collisions at intermediate energies. It is shown that the nucleon transverse flow in heavy-ion collisions especially those involving polarized projectile and/or target nuclei depends strongly on the spin-orbit coupling. Future comparisons of model simulations with experimental data will help constrain the poorly known in-medium nucleon spin-orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.
Spinor Boltzmann Equation with Two Momenta at the Fermi Level
Institute of Scientific and Technical Information of China (English)
王正川
2012-01-01
Based on the formalism of Keldysh's nonequilibrium Green function, we establish a two momenta spinor Boltzmann equation for longitudinal scalar distribution function and transverse vector distribution function. The lon- gitudinal charge currents, transverse spin currents and the continuity equations satisfied by them are then studied, it indicates that both the charge currents and spin currents decay oscillately along with position, which is due to the momenta integral over the Fermi surface. We also compare our charge currents and spin currents with the corresponding results of one momentum spinor Boltzmann equation, the differences are obvious.
Generalized Relativistic Chapman-Enskog Solution of the Boltzmann Equation
García-Perciante, A L; García-Colin, L S
2007-01-01
The Chapman-Enskog method of solution of the relativistic Boltzmann equation is generalized in order to admit a time-derivative term associated to the thermodynamic force in its first order solution. Both existence and uniqueness of such a solution are proved based on the standard theory of integral equations. The mathematical implications of the generalization here introduced are briefly explored.
Celebrating Cercignani's conjecture for the Boltzmann equation
Desvillettes, Laurent; Villani, Cédric
2010-01-01
Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.
Acoustic limit of the Boltzmann equation: classical solutions
Jang, Juhi; Jiang, Ning
2009-01-01
We study the acoustic limit from the Boltzmann equation in the framework of classical solutions. For a solution $F_\\varepsilon=\\mu +\\varepsilon \\sqrt{\\mu}f_\\varepsilon$ to the rescaled Boltzmann equation in the acoustic time scaling \\partial_t F_\\varepsilon +\\vgrad F_\\varepsilon =\\frac{1}{\\varepsilon} \\Q(F_\\varepsilon,F_\\varepsilon), inside a periodic box $\\mathbb{T}^3$, we establish the global-in-time uniform energy estimates of $f_\\varepsilon$ in $\\varepsilon$ and prove that $f_\\varepsilon$...
Axisymmetric multiphase Lattice Boltzmann method for generic equations of state
Reijers, Sten A; Toschi, Federico
2015-01-01
We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid-gas density ratios up to $10^3$. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation equations. We validate the model by showing that a stationary droplet obeys the Young-Laplace law, comparing the second oscillation mode of a droplet with respect to an analytical solution and showing correct mass conservation of a propagating density wave.
Numerical solution of Boltzmann's equation
Energy Technology Data Exchange (ETDEWEB)
Sod, G.A.
1976-04-01
The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig.
Metamaterial characterization using Boltzmann's kinetic equation for electrons
DEFF Research Database (Denmark)
Novitsky, Andrey; Zhukovsky, Sergei; Novitsky, D.;
2013-01-01
Statistical properties of electrons in metals are taken into consideration to describe the microscopic motion of electrons. Assuming degenerate electron gas in metal, we introduce the Boltzmann kinetic equation to supplement Maxwell's equations. The solution of these equations clearly shows the r...... the resonant behavior of electronic response to an external electromagnetic field. We demonstrate the approach for planar and circular geometries of the metamolecules....
Thermal creep problems by the discrete Boltzmann equation
Directory of Open Access Journals (Sweden)
L. Preziosi
1991-05-01
Full Text Available This paper deals with an initial-boundary value problem for the discrete Boltzmann equation confined between two moving walls at different temperature. A model suitable for the quantitative analysis of the initial boundary value problem and the relative existence theorem are given.
CORRECTIONS TO THE COLLISION TERM IN THE BGK BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; REN RONG-CAI; CUI XIAO-PENG; JI ZHONG-ZHEN
2001-01-01
With the discrete method of the hexagonal cell and three different velocities of particle population in each cell,a two-dimensional lattice Boltzmann model is developed in this paper.[1,2] The collision operator in the Boltzmann equation is expanded to fourth order using the Taylor expansion.[3,4] With this model, good results have been obtained from the numerical simulation of the reflection phenomenon of the shock wave on the surface of an obstacle, and the numerical stability is also good. Thus the applicability of the D2Q19 model is verified.
The Nonclassical Diffusion Approximation to the Nonclassical Linear Boltzmann Equation
Vasques, Richard
2015-01-01
We show that, by correctly selecting the probability distribution function $p(s)$ for a particle's distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of $p(s)$ preserves the $true$ mean-squared free path of the system, which sheds new light on the results obtained in previous work.
Green, B. I.; Vedula, Prakash
2013-07-01
An alternative approach for solution of the collisional Boltzmann equation for a lattice architecture is presented. In the proposed method, termed the collisional lattice Boltzmann method (cLBM), the effects of spatial transport are accounted for via a streaming operator, using a lattice framework, and the effects of detailed collisional interactions are accounted for using the full collision operator of the Boltzmann equation. The latter feature is in contrast to the conventional lattice Boltzmann methods (LBMs) where collisional interactions are modeled via simple equilibrium based relaxation models (e.g. BGK). The underlying distribution function is represented using weights and fixed velocity abscissas according to the lattice structure. These weights are evolved based on constraints on the evolution of generalized moments of velocity according to the collisional Boltzmann equation. It can be shown that the collision integral can be reduced to a summation of elementary integrals, which can be analytically evaluated. The proposed method is validated using studies of canonical microchannel Couette and Poiseuille flows (both body force and pressure driven) and the results are found to be in good agreement with those obtained from conventional LBMs and experiments where available. Unlike conventional LBMs, the proposed method does not involve any equilibrium based approximations and hence can be useful for simulation of highly nonequilibrium flows (for a range of Knudsen numbers) using a lattice framework.
Analysis of Jeans instability from the Boltzmann equation
Kremer, Gilberto M.
2016-11-01
The dynamics of self-gravitating fluids is analyzed within the framework of a collisionless Boltzmann equation in the presence of gravitational fields and Poisson equation. Two cases are analyzed: a system with baryonic and dark matter in a static universe and a single system in an expanding universe. The amplitudes of the perturbed distribution functions are considered as a linear combination of the collision invariants of the Boltzmann equation. For the system of baryonic and dark matter, the Jeans mass of the combined system is smaller than the one of the single system indicating that a smaller mass is needed to initiate the collapse. For the single system in an expanding universe it is not necessary to make use of Jeans "swindle"and it shown that for small wavelengths the density contrast oscillates while for large wavelengths it grows with time and the Jeans instability emerges.
CMB spectral distortions as solutions to the Boltzmann equations
Ota, Atsuhisa
2016-01-01
We newly re-interpret cosmic microwave background spectral distortions as solutions to the Boltzmann equation at second order. This approach makes it possible to solve the equation of the momentum dependent temperature perturbations explicitly. In addition, we define higher order spectral distortions systematically, assuming that the collision term is linear in the photon distribution functions. For example, we find the linear Sunyaev-Zel'dovich effect whose momentum shape is different from the usual $y$ distortion, and show that the higher order spectral distortions are also generated as a result of the diffusion process in a language of higher order Boltzmann equations. The method may be applicable to a wider class of problems and has potential to give a general prescription to non-equilibrium physics.
Analytical solution of the Boltzmann-Poisson equation and its application to MIS tunneling junctions
Institute of Scientific and Technical Information of China (English)
Zhang Li-Zhi; Wang Zheng-Chuan
2009-01-01
In order to consider quantum transport under the influence of an electron-electron (e-e) interaction in a mesoscopic conductor, the Boltzmann equation and Poisson equation are investigated jointly. The analytical expressions of the distribution function for the Boltzmann equation and the self-consistent average potential concerned with e-e interaction are obtained, and the dielectric function appearing in the self-consistent average potential is naturally generalized beyond the Thomas-Fermi approximation. Then we apply these results to the tunneling junctions of a metal-insulatorsemiconductor (MIS) in which the electrons are accumulated near the interface of the semiconductor, and we find that the e-e interaction plays an important role in the transport procedure of this system. The electronic density, electric current as well as screening Coulombic potential in this case are studied, and we reveal the time and position dependence of these physical quantities explicitly affected by the e-e interaction.
Asymptotic stability of the Boltzmann equation with Maxwell boundary conditions
Briant, Marc; Guo, Yan
2016-12-01
In a general C1 domain, we study the perturbative Cauchy theory for the Boltzmann equation with Maxwell boundary conditions with an accommodation coefficient α in (√{ 2 / 3 } , 1 ], and discuss this threshold. We consider polynomial or stretched exponential weights m (v) and prove existence, uniqueness and exponential trend to equilibrium around a global Maxwellian in Lx,v∞ (m). Of important note is the fact that the methods do not involve contradiction arguments.
A Boltzmann Transport Simulation Using Open Source Physics
Hasbun, Javier
2004-03-01
The speed of a charged particle, under an applied electric field, in a conducting media, is, usually, simply modelled by writing Newton's 2nd law in the form mfrac ddtv=qE-mfrac vτ ; (1), where v is the speed, E is the applied electric field, q is the charge, m is the mass, and τ is the scattering time between collisions. Here, we simulate a numerical solution of the Boltzmann transport equation,frac partial partial tf+ vot nabla _rf+Fot nabla _pf=frac partial partial tf|_coll (2), where in general the Boltzmann distribution function f=f(r,p,t) depends on position, momentum, and time. Our numerical solution is made possible by neglecting the 2nd term on the LHS, and by modelling the RHS collision term as fracpartial partial tf|_coll=-frac 1τ . With these approximations, in addition to considering only one dimension, we find, our numerical solution of (2). The average velocity numerically obtained through the resulting distribution is compared to that obtained by the analytic solution of (1). An efficient method of carrying out the numerical solution of (2) due to P. Drallos and M. Wadehra [Journal of Applied Physics 63, 5601(1988)] is incorporated here. A final version of an applet that performs the full Java simulation will be located at http://www.westga.edu/ jhasbun/osp/osp.htm.
LATTICE BOLTZMANN EQUATION MODEL IN THE CORIOLIS FIELD
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; MAO JIANG-YU; ZHANG QIONG
2001-01-01
In a large-scale field of rotational fluid, various unintelligible and surprising dynamic phenomena are produced due to the effect of the Coriolis force. The lattice Boltzmann equation (LBE) model in the Coriolis field is developed based on previous works.[1-4] Geophysical fluid dynamics equations are derived from the model. Numerical simulations have been made on an ideal atmospheric circulation of the Northern Hemisphere by using the model and they reproduce the Rossby wave motion well. Hence the applicability of the model is verified in both theory and experiment.
Nonequilibrium Spin Magnetization Quantum Transport Equations
Buot, F A; Otadoy, R E S; Villarin, D L
2011-01-01
The classical Bloch equations of spin magnetization transport is extended to fully time-dependent and highly-nonlinear nonequilibrium quantum distribution function (QDF) transport equations. The leading terms consist of the Boltzmann kinetic equation with spin-orbit coupling in a magnetic field together with spin-dependent scattering terms which do not have any classical analogue, but should incorporate the spatio-temporal-dependent phase-space dynamics of Elliot-Yafet and D'yakonov-Perel scatterings. The resulting magnetization QDF transport equation serves as a foundation for computational spintronic and nanomagnetic device applications, in performing simulation of ultrafast-switching-speed/low-power performance and reliability analyses.
A new lattice Boltzmann equation to simulate density-driven convection of carbon dioxide
Allen, Rebecca
2013-01-01
The storage of CO2 in fluid-filled geological formations has been carried out for more than a decade in locations around the world. After CO2 has been injected into the aquifer and has moved laterally under the aquifer\\'s cap-rock, density-driven convection becomes an important transport process to model. However, the challenge lies in simulating this transport process accurately with high spatial resolution and low CPU cost. This issue can be addressed by using the lattice Boltzmann equation (LBE) to formulate a model for a similar scenario when a solute diffuses into a fluid and density differences lead to convective mixing. The LBE is a promising alternative to the traditional methods of computational fluid dynamics. Rather than discretizing the system of partial differential equations of classical continuum mechanics directly, the LBE is derived from a velocity-space truncation of the Boltzmann equation of classical kinetic theory. We propose an extension to the LBE, which can accurately predict the transport of dissolved CO2 in water, as a step towards fluid-filled porous media simulations. This is achieved by coupling two LBEs, one for the fluid flow and one for the convection and diffusion of CO2. Unlike existing lattice Boltzmann equations for porous media flow, our model is derived from a system of moment equations and a Crank-Nicolson discretization of the velocity-truncated Boltzmann equation. The forcing terms are updated locally without the need for additional central difference approximation. Therefore our model preserves all the computational advantages of the single-phase lattice Boltzmann equation and is formally second-order accurate in both space and time. Our new model also features a novel implementation of boundary conditions, which is simple to implement and does not suffer from the grid-dependent error that is present in the standard "bounce-back" condition. The significance of using the LBE in this work lies in the ability to efficiently
Phonon Boltzmann equation-based discrete unified gas kinetic scheme for multiscale heat transfer
Guo, Zhaoli
2016-01-01
Numerical prediction of multiscale heat transfer is a challenging problem due to the wide range of time and length scales involved. In this work a discrete unified gas kinetic scheme (DUGKS) is developed for heat transfer in materials with different acoustic thickness based on the phonon Boltzmann equation. With discrete phonon direction, the Boltzmann equation is discretized with a second-order finite-volume formulation, in which the time-step is fully determined by the Courant-Friedrichs-Lewy (CFL) condition. The scheme has the asymptotic preserving (AP) properties for both diffusive and ballistic regimes, and can present accurate solutions in the whole transition regime as well. The DUGKS is a self-adaptive multiscale method for the capturing of local transport process. Numerical tests for both heat transfers with different Knudsen numbers are presented to validate the current method.
Asinari, P.
2011-03-01
Boltzmann equation is one the most powerful paradigms for explaining transport phenomena in fluids. Since early fifties, it received a lot of attention due to aerodynamic requirements for high altitude vehicles, vacuum technology requirements and nowadays, micro-electro-mechanical systems (MEMs). Because of the intrinsic mathematical complexity of the problem, Boltzmann himself started his work by considering first the case when the distribution function does not depend on space (homogeneous case), but only on time and the magnitude of the molecular velocity (isotropic collisional integral). The interest with regards to the homogeneous isotropic Boltzmann equation goes beyond simple dilute gases. In the so-called econophysics, a Boltzmann type model is sometimes introduced for studying the distribution of wealth in a simple market. Another recent application of the homogeneous isotropic Boltzmann equation is given by opinion formation modeling in quantitative sociology, also called socio-dynamics or sociophysics. The present work [1] aims to improve the deterministic method for solving homogenous isotropic Boltzmann equation proposed by Aristov [2] by two ideas: (a) the homogeneous isotropic problem is reformulated first in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium).
Generalizing the Boltzmann equation in complex phase space
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014), 10.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015), 10.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others.
Energy Technology Data Exchange (ETDEWEB)
Zabadal, Jorge; Borges, Volnei; Van der Laan, Flavio T., E-mail: jorge.zabadal@ufrgs.br, E-mail: borges@ufrgs.br, E-mail: ftvdl@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Engenharia Mecanica. Grupo de Pesquisas Radiologicas; Ribeiro, Vinicius G., E-mail: vinicius_ribeiro@uniritter.edu.br [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio G., E-mail: phd.marcio@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Tramandai, RS (Brazil). Departamento Interdisciplinar do Campus Litoral Norte
2015-07-01
This work presents a new analytical method for solving the Boltzmann equation. In this formulation, a linear differential operator is applied over the Boltzmann model, in order to produce a partial differential equation in which the scattering term is absent. This auxiliary equation is solved via reduction of order. The exact solution obtained is employed to define a precursor for the buildup factor. (author)
Nonaligned shocks for discrete velocity models of the Boltzmann equation
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J. M. Greenberg
1991-05-01
Full Text Available At the conclusion of I. Bonzani's presentation on the existence of structured shock solutions to the six-velocity, planar, discrete Boltzmann equation (with binary and triple collisions, Greenberg asked whether such solutions were possible in directions e(α=(cosα ,sinα when α was not one of the particle flow directions. This question generated a spirited discussion but the question was still open at the conclusion of the conference. In this note the author will provide a partial resolution to the question raised above. Using formal perturbation arguments he will produce approximate solutions to the equation considered by Bonzani which represent traveling waves propagating in any direction e(α=(cosα ,sinα.
Interplay of Boltzmann equation and continuity equation for accelerated electrons in solar flares
Codispoti, Anna
2015-01-01
During solar flares a large amount of electrons are accelerated within the plasma present in the solar atmosphere. Accurate measurements of the motion of these electrons start becoming available from the analysis of hard X-ray imaging-spectroscopy observations. In this paper, we discuss the linearized perturbations of the Boltzmann kinetic equation describing an ensemble of electrons accelerated by the energy release occurring during solar flares. Either in the limit of high energy or at vanishing background temperature such an equation reduces to a continuity equation equipped with an extra force of stochastic nature. This stochastic force is actually described by the well known energy loss rate due to Coulomb collision with ambient particles, but, in order to match the collision kernel in the linearized Boltzmann equation it needs to be treated in a very specific manner. In the second part of the paper the derived continuity equation is solved with some hyperbolic techniques, and the obtained solution is wr...
Flavoured quantum Boltzmann equations from cQPA
Fidler, Christian; Kainulainen, Kimmo; Rahkila, Pyry Matti
2011-01-01
We develop a Boltzmann-type quantum transport theory for interacting fermion and scalar fields including both flavour and particle-antiparticle mixing. Our formalism is based on the coherent quasiparticle approximation (cQPA) for the 2-point correlation functions, whose extended phase-space structure contains new spectral shells for flavour- and particle-antiparticle coherence. We derive explicit cQPA propagators and Feynman rules for the transport theory. In particular the nontrivial Wightman functions can be written as composite operators $\\sim {\\cal A} F {\\cal A}$, which generalize the usual Kadanoff-Baym ansatz. Our numerical results show that particle-antiparticle coherence can strongly influence CP-violating flavour mixing even for relatively slowly-varying backgrounds. Thus, unlike recently suggested, these correlations cannot be neglected when studying asymmetry generation due to time-varying mass transition, for example in electroweak-type baryogenesis models. Finally, we show that the cQPA coherence...
Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation
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Bertrand Lods
2015-06-01
Full Text Available Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set Ɛ of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if Ɛ is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell’s weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyvärinen divergence. This requires us to generalize our approach to Orlicz–Sobolev spaces to include derivatives.
Acoustic equations of state for simple lattice Boltzmann velocity sets.
Viggen, Erlend Magnus
2014-07-01
The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ξ(iα)∈(-1,0,1) for all i, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.
An Exact Solution to the Two-Particle Boltzmann Equation System for Maxwell Gases
Institute of Scientific and Technical Information of China (English)
布仁满都拉; 赵迎春
2012-01-01
An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.
Discretization of the velocity space in solution of the Boltzmann equation
Shan, X; Shan, Xiaowen; He, Xiaoyi
1998-01-01
We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to be equivalent to truncating the Hermite expansion of the distribution function to the corresponding order. The truncated part of the distribution has no contribution to the moments of low orders and is negligible at small Mach numbers. Higher order approximations to the Boltzmann equation can be achieved by using more velocities in the quadrature.
On kinetic Boltzmann equations and related hydrodynamic flows with dry viscosity
Directory of Open Access Journals (Sweden)
Nikolai N. Bogoliubov (Jr.
2007-01-01
Full Text Available A two-component particle model of Boltzmann-Vlasov type kinetic equations in the form of special nonlinear integro-differential hydrodynamic systems on an infinite-dimensional functional manifold is discussed. We show that such systems are naturally connected with the nonlinear kinetic Boltzmann-Vlasov equations for some one-dimensional particle flows with pointwise interaction potential between particles. A new type of hydrodynamic two-component Benney equations is constructed and their Hamiltonian structure is analyzed.
A Lattice Boltzmann Model and Simulation of KdV-Burgers Equation
Institute of Scientific and Technical Information of China (English)
ZHANGChao-Ying; TANHui-Li; LIUMu-Ren; KONGLing-Jiang
2004-01-01
A lattice Boltzmann model of KdV-Burgers equation is derived by using the single-relaxation form of the lattice Boltzmann equation. With the present model, we simulate the traveling-wave solutions, the solitary-wave solutions, and the sock-wave solutions of KdV-Burgers equation, and calculate the decay factor and the wavelength of the sock-wave solution, which has exponential decay. The numerical results agree with the analytical solutions quite well.
A New Lattice Boltzmann Model for KdV-Burgers Equation
Institute of Scientific and Technical Information of China (English)
MA Chang-Feng
2005-01-01
@@ A new lattice Boltzmann model with amending-function for KdV-Burgers equation, ut +uux - αuxx +βuxxx = 0,is presented by using the single-relaxation form of the lattice Boltzmann equation. Applying the proposed model,we simulate the solutions ofa kind of KdV-Burgers equations, and the numerical results agree with the analytical solutions quite well.
Distributional Monte Carlo Methods for the Boltzmann Equation
2013-03-01
become the first to possess non - Maxwellian distributions, and therefore become the only location where 112 collisions are required to be calculated... Maxwellian . . . . . . . . . . . . . . . . . 16 fMB Maxwell-Boltzmann Density . . . . . . . . . . . . . . . . . . . . . . . . 16 nMB Maxwell-Boltzmann...is equivalent to assuming that millions of actual particles all share the exact velocity vector. This assumption is non -physical in the sense that
Boltzmann Transport in Hybrid PIC HET Modeling
2015-07-01
Symposium on Space Technology and Science, 34th International Electric Propulsion Conference and 6th Nano-satellite Symposium Hyogo-Kobe, Japan July 4–10...key source of uncertainty inhibiting first-principal simulation of Hall -effect thruster (HET) physics. Fluid electron models depend heavily on the...transport models are a key source of uncertainty inhibiting first- principal simulation of Hall -effect thruster (HET) physics. Fluid electron models
A Stability Notion for the viscous Shallow Water Lattice Boltzmann Equations
Banda, Mapundi K
2015-01-01
The stability of Lattice Boltzmann Equations modelling Shallow Water Equations in the special case of reduced gravity is investigated theoretically. A stability notion is defined as applied in incompressible Navier-Stokes equations in Banda, M. K., Yong, W.- A. and Klar, A: A stability notion for lattice Boltzmann equations. SIAM J. Sci. Comput. {\\bf 27(6)}, 2098-2111 (2006). It is found that to maintain stability a careful choice of the value of the reduced gravity must be made. The stability notion is employed to investigate different shallow water lattice Boltzmann Equations. The effect of the reduced gravity on the mechanism of instability is investigated. Results are tested using the Lattice Boltzmann Method for various values of the governing parameters of the flow. It is observed that even for the discrete model the reduced gravity has a significant effect on the stability.
Electronic transport in VO{sub 2}—Experimentally calibrated Boltzmann transport modeling
Energy Technology Data Exchange (ETDEWEB)
Kinaci, Alper; Rosenmann, Daniel; Chan, Maria K. Y., E-mail: debasish.banerjee@toyota.com, E-mail: mchan@anl.gov [Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439 (United States); Kado, Motohisa [Higashifuji Technical Center, Toyota Motor Corporation, Susono, Shizuoka 410-1193 (Japan); Ling, Chen; Zhu, Gaohua; Banerjee, Debasish, E-mail: debasish.banerjee@toyota.com, E-mail: mchan@anl.gov [Materials Research Department, Toyota Motor Engineering and Manufacturing North America, Inc., Ann Arbor, Michigan 48105 (United States)
2015-12-28
Materials that undergo metal-insulator transitions (MITs) are under intense study, because the transition is scientifically fascinating and technologically promising for various applications. Among these materials, VO{sub 2} has served as a prototype due to its favorable transition temperature. While the physical underpinnings of the transition have been heavily investigated experimentally and computationally, quantitative modeling of electronic transport in the two phases has yet to be undertaken. In this work, we establish a density-functional-theory (DFT)-based approach with Hubbard U correction (DFT + U) to model electronic transport properties in VO{sub 2} in the semiconducting and metallic regimes, focusing on band transport using the Boltzmann transport equations. We synthesized high quality VO{sub 2} films and measured the transport quantities across the transition, in order to calibrate the free parameters in the model. We find that the experimental calibration of the Hubbard correction term can efficiently and adequately model the metallic and semiconducting phases, allowing for further computational design of MIT materials for desirable transport properties.
Equivalence Between Forward and Backward Boltzmann Equations in Multi-Component Medium
Institute of Scientific and Technical Information of China (English)
张竹林
2002-01-01
The author generalized the propagator function theory introduced first by Sigmund, and gave a explicitly proof of a equivalence between forward and backward Boltzmann equations in a multi-component medium by using the generalized propagator function theory.
Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity
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Shi-you Lin
2014-01-01
Full Text Available The Cauchy problem of the nonlinear spatially homogeneous Boltzmann equation without angular cutoff is studied. By using analytic techniques, one proves the Gevrey regularity of the C∞ solutions in non-Maxwellian and strong singularity cases.
Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation Project
National Aeronautics and Space Administration — The overall objective of the proposed project is to develop a generalized lattice Boltzmann (GLB) approach as a potential computational aeroacoustics (CAA) tool for...
Note on Invariance of One-Dimensional Lattice-Boltzmann Equation
Institute of Scientific and Technical Information of China (English)
RAN Zheng
2007-01-01
Invariance of the one-dimensional lattice Boltzmann model is proposed together with its rigorous theoretical background.It is demonstrated that the symmetry inherent in Navier-Stokes equations is not really recovered in the one-dimensional lattice Boltzmann equation (LBE),especially for shock calculation.Symmetry breaking may be the inherent cause for the non-physical oscillations in the vicinity of the shock for LBE calculation.
THE NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE IN THE WHOLE SPACE
Institute of Scientific and Technical Information of China (English)
Yuanjie LEI
2014-01-01
This paper is concerned with the non-cutoff Boltzmann equation for full-range interactions with potential force in the whole space. We establish the global existence and optimal temporal convergence rates of classical solutions to the Cauchy problem when initial data is a small perturbation of the stationary solution. The analysis is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [1-3, 15] and the non-cutoff Vlasov-Poisson-Boltzmann system [6].
Jet propagation within a Linearized Boltzmann Transport model
Energy Technology Data Exchange (ETDEWEB)
Luo, Tan; He, Yayun [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Wang, Xin-Nian [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Nuclear Science Division, Mailstop 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94740 (United States); Zhu, Yan [Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, E-15706 Santiago de Compostela, Galicia (Spain)
2014-12-15
A Linearized Boltzmann Transport (LBT) model has been developed for the study of parton propagation inside quark–gluon plasma. Both leading and thermal recoiled partons are tracked in order to include the effect of jet-induced medium excitation. In this talk, we present a study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons and jet-induced medium excitations are found to have significant influences on the jet energy loss and transverse profile.
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong
2016-06-01
Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier-Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region
A Stability notion for the viscous Shallow Water Discrete-Velocity Boltzmann Equations
Banda, Mapundi K.; Uoane, Tumelo R. A.
2015-01-01
The stability of Lattice Boltzmann Equations modelling Shallow Water Equations in the special case of reduced gravity is investigated theoretically. A stability notion is defined as applied in incompressible Navier-Stokes equations in Banda, M. K., Yong, W.- A. and Klar, A: A stability notion for lattice Boltzmann equations. SIAM J. Sci. Comput. {\\bf 27(6)}, 2098-2111 (2006). It is found that to maintain stability a careful choice of the value of the reduced gravity must be made. The stabilit...
Lattice Boltzmann Model for The Volume-Averaged Navier-Stokes Equations
Zhang, Jingfeng; Ouyang, Jie
2014-01-01
A numerical method, based on discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model can accurately reproduce both the linear and nonlinear drag effects of porosity in the fluid flow through porous media.
Wu, Lei; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2015-08-01
The Boltzmann equation with an arbitrary intermolecular potential is solved by the fast spectral method. As examples, noble gases described by the Lennard-Jones potential are considered. The accuracy of the method is assessed by comparing both transport coefficients with variational solutions and mass/heat flow rates in Poiseuille/thermal transpiration flows with results from the discrete velocity method. The fast spectral method is then applied to Fourier and Couette flows between two parallel plates, and the influence of the intermolecular potential on various flow properties is investigated. It is found that for gas flows with the same rarefaction parameter, differences in the heat flux in Fourier flow and the shear stress in Couette flow are small. However, differences in other quantities such as density, temperature, and velocity can be very large.
Bazow, D; Heinz, U; Martinez, M; Noronha, J
2016-01-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lema\\^itre-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of non-hydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the non-hydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation...
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Nilson C. Roberty
2011-01-01
Full Text Available We introduce algorithms marching over a polygonal mesh with elements consistent with the propagation directions of the particle (radiation flux. The decision for adopting this kind of mesh to solve the one-speed Boltzmann transport equation is due to characteristics of the domain of the transport operator which controls derivatives only in the direction of propagation of the particles (radiation flux in the absorbing and scattering media. This a priori adaptivity has the advantages that it formulates a consistent scheme which makes appropriate the application of the Lax equivalence theorem framework to the problem. In this work, we present the main functional spaces involved in the formalism and a description of the algorithms for the mesh generation and the transport equation solution. Some numerical examples related to the solution of a transmission problem in a high-contrast model with absorption and scattering are presented. Also, a comparison with benchmarks problems for source and reactor criticality simulations shows the compatibility between calculations with the algorithms proposed here and theoretical results.
Peristaltic particle transport using the Lattice Boltzmann method
Energy Technology Data Exchange (ETDEWEB)
Connington, Kevin William [Los Alamos National Laboratory; Kang, Qinjun [Los Alamos National Laboratory; Viswanathan, Hari S [Los Alamos National Laboratory; Abdel-fattah, Amr [Los Alamos National Laboratory; Chen, Shiyi [JOHNS HOPKINS UNIV.
2009-01-01
Peristaltic transport refers to a class of internal fluid flows where the periodic deformation of flexible containing walls elicits a non-negligible fluid motion. It is a mechanism used to transport fluid and immersed solid particles in a tube or channel when it is ineffective or impossible to impose a favorable pressure gradient or desirous to avoid contact between the transported mixture and mechanical moving parts. Peristaltic transport occurs in many physiological situations and has myriad industrial applications. We focus our study on the peristaltic transport of a macroscopic particle in a two-dimensional channel using the lattice Boltzmann method. We systematically investigate the effect of variation of the relevant dimensionless parameters of the system on the particle transport. We find, among other results, a case where an increase in Reynolds number can actually lead to a slight increase in particle transport, and a case where, as the wall deformation increases, the motion of the particle becomes non-negative only. We examine the particle behavior when the system exhibits the peculiar phenomenon of fluid trapping. Under these circumstances, the particle may itself become trapped where it is subsequently transported at the wave speed, which is the maximum possible transport in the absence of a favorable pressure gradient. Finally, we analyze how the particle presence affects stress, pressure, and dissipation in the fluid in hopes of determining preferred working conditions for peristaltic transport of shear-sensitive particles. We find that the levels of shear stress are most hazardous near the throat of the channel. We advise that shear-sensitive particles should be transported under conditions where trapping occurs as the particle is typically situated in a region of innocuous shear stress levels.
Mahakrishnan, Sathiya; Chakraborty, Subrata; Vijay, Amrendra
2016-09-15
Diffusion, an emergent nonequilibrium transport phenomenon, is a nontrivial manifestation of the correlation between the microscopic dynamics of individual molecules and their statistical behavior observed in experiments. We present a thorough investigation of this viewpoint using the mathematical tools of quantum scattering, within the framework of Boltzmann transport theory. In particular, we ask: (a) How and when does a normal diffusive transport become anomalous? (b) What physical attribute of the system is conceptually useful to faithfully rationalize large variations in the coefficient of normal diffusion, observed particularly within the dynamical environment of biological cells? To characterize the diffusive transport, we introduce, analogous to continuous phase transitions, the curvature of the mean square displacement as an order parameter and use the notion of quantum scattering length, which measures the effective interactions between the diffusing molecules and the surrounding, to define a tuning variable, η. We show that the curvature signature conveniently differentiates the normal diffusion regime from the superdiffusion and subdiffusion regimes and the critical point, η = ηc, unambiguously determines the coefficient of normal diffusion. To solve the Boltzmann equation analytically, we use a quantum mechanical expression for the scattering amplitude in the Boltzmann collision term and obtain a general expression for the effective linear collision operator, useful for a variety of transport studies. We also demonstrate that the scattering length is a useful dynamical characteristic to rationalize experimental observations on diffusive transport in complex systems. We assess the numerical accuracy of the present work with representative experimental results on diffusion processes in biological systems. Furthermore, we advance the idea of temperature-dependent effective voltage (of the order of 1 μV or less in a biological environment, for example
Discrete Boltzmann model of shallow water equations with polynomial equilibria
Meng, Jianping; Emerson, David R; Peng, Yong; Zhang, Jianmin
2016-01-01
A hierarchy of discrete Boltzmann model is proposed for simulating shallow water flows. By using the Hermite expansion and Gauss-Hermite quadrature, the conservation laws are automatically satisfied without extra effort. Moreover, the expansion order and quadrature can be chosen flexibly according to the problem for striking the balance of accuracy and efficiency. The models are then tested using the classical one-dimensional dam-breaking problem, and successes are found for both supercritical and subcritical flows.
A New Regularization Mechanism for the Boltzmann Equation Without Cut-Off
Silvestre, Luis
2016-11-01
We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Hölder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in {L^∞} which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded.
d'Eon, Eugene
2013-01-01
We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation (GLBE) for the stationary collision density and scalar flux about an isotropic point source in an infinite $d$-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentially-distributed free paths in arbitrary dimensions as well as a number of non-classical transport theories (non-exponential random flights) that describe a broader class of transport processes within partially-correlated random media. New rigorous asymptotic diffusion approximations are derived where possible. We also generalize Grosjean's moment-preserving approach of separating the first (or uncollided) distribution from the collided portion and approximating only the latter using diffusion. We find that for any spatial dimension and for many free-path distributions Grosjean's approach produces compact, analytic approximations that are, overall, more accurate for high absorption and f...
Chen, Li; Zhang, Lei; Tao, Wenquan
2014-01-01
Size, morphology and distributions of pores in organic matters of shale matrix are discussed based on high resolution images from experiments in the literature. 150 nanoscale structures of the organic matters are then reconstructed by randomly placing pore spheres with different diameters and overlap tolerances. Effects of porosity, the mean diameter and the overlap tolerance on void space connectivity and pore size distribution are studied. Further, a pore-scale model based on the Lattice Boltzmann method is developed to predict the Knudsen diffusivity and permeability of the reconstructed organic matters. The simulation results show that the mean pore diameter and overlap tolerance significantly affect the transport properties. The predicted Knudsen effective diffusivity is compared with Bruggeman equation and it is found that this equation underestimate the tortuosity. A modified Bruggeman equation is proposed based on the simulation results. The predicted intrinsic permeability is in acceptable agreement ...
An H Theorem for Boltzmann's Equation for the Yard-Sale Model of Asset Exchange
Boghosian, Bruce M.; Johnson, Merek; Marcq, Jeremy A.
2015-12-01
In recent work (Boghosian, Phys Rev E 89:042804-042825, 2014; Boghosian, Int J Mod Phys 25:1441008-1441015, 2014), Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an H function of both the Boltzmann and Fokker-Planck equations for the model.
Kinetic Description for a Suspension of Inelastic Spheres - Boltzmann and BGK Equations
2007-11-02
Kinetic description for a suspension of inelastic spheres - Boltzmann and BGK equations Cedric Croizet and Renee Gatignol Laboratoire de Modelisation ...Organization Name(s) and Address(es) Laboratoire de Modelisation en Mecanique - Universite Pierre et Marie Curie (Paris 6) et CNRS UMR 7607 - 4) place
Convergence Rate to Stationary Solutions for Boltzmann Equation with External Force
Institute of Scientific and Technical Information of China (English)
Seiji UKAI; Tong YANG; Huijiang ZHAO
2006-01-01
For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. (2005) through the energy method. Based on this stability analysis and some techniques on analyzing the convergence rates to stationary solutions for the compressible Navier-Stokes equations, in this paper, we study the convergence rate to the above stationary solutions for the Boltzmann equation which is a fundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components and the dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained by constructing some energy functionals.
Serov, S A
2013-01-01
In the article correct method for the kinetic Boltzmann equation asymptotic solution is formulated, the Hilbert's and Enskog's methods are discussed. The equations system of multicomponent non-equilibrium gas dynamics is derived, that corresponds to the first order in the approximate (asymptotic) method for solution of the system of kinetic Boltzmann equations. It is shown, that the velocity distribution functions of particles, obtained by the proposed method and by Enskog's method, within Enskog's approach, are equivalent up to the infinitesimal first order terms of the asymptotic expansion, but, generally speaking, differ in the next order. Interpretation of turbulent gas flow is proposed, as stratified on components gas flow, which is described by the derived equations system of multicomponent non-equilibrium gas dynamics.
A unified lattice Boltzmann model for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Chai Zhenhua [State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074 (China); Shi Baochang [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)], E-mail: sbchust@126.com; Zheng Lin [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)
2008-05-15
In this paper, a unified and novel lattice Boltzmann model is proposed for solving nonlinear partial differential equation that has the form DU{sub t} + {alpha}UU{sub x} + {beta}U{sup n}U{sub x} - {gamma}U{sub xx} + {delta} U{sub xxx} = F(x,t). Numerical results agree well with the analytical solutions and results derived by existing literature, which indicates the present model is satisfactory and efficient on solving nonlinear partial differential equations.
Steady detonation waves via the Boltzmann equation for a reacting mixture
Conforto, F; Schürrer, F; Ziegler, I
2003-01-01
Based on the Boltzmann equation, the detonation problem is dealt with on a mesoscopic level. The model is based on the assumption that ahead of a shock an explosive gas mixture is in meta stable equilibrium. Starting from the Von Neumann point the chemical reaction, initiated by the pressure jump, proceeds until the chemical equilibrium is reached. Numerical solutions of the derived macroscopic equations as well as the corresponding Hugoniot diagrams which reveal the physical relevance of the mathematical model are provided.
Generalized Poisson—Boltzmann Equation Taking into Account Ionic Interaction and Steric Effects
Liu, Xin-Min; Li, Hang; Li, Rui; Tian, Rui; Xu, Chen-Yang
2012-09-01
Generalized Poisson—Boltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density 0.15 mol/l in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.
Electric-field conditions for Landauer and Boltzmann-Drude conductance equations
Fenton, E. W.
1992-08-01
It is shown explicitly in a unified theory of conductance, for bulk metals and mesoscopic systems, that a Landauer type of conductance equation is compatible with a spatially localized continuous-q-spectrum electric field that is unidirectional, but not with a homogeneous q=0 field. The reverse field condition holds for the Boltzmann-Drude conductance equation for an inhomogeneous bulk metal that has no inelastic scattering. A Feynman-diagram form of Green-function theory shows explicitly the virtual processes and repeated quantum scattering from a single object that occur with Feynman path integrals. The distinction between repeated scattering of current and repeated one-electron scattering is important. For a mesoscopic system, infinite conduction would occur if scattering were to be exactly zero-there is no necessity for postulated contact potentials between lead wires and thermal reservoirs. This is because just in this translationally invariant case a q=0 electric field must occur, and for this the Landauer equation must be replaced by the Boltzmann-Drude equation with zero scattering. In contrast to the strong frequency dependence of the Boltzmann-Drude equation, it is shown that no frequency dependence of the conductance occurs in the Landauer type of equation for frequencies much smaller than the inverse of the electron transit time across the electric-field region.
The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium
Briant, Marc; Daus, Esther S.
2016-12-01
We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in {L^1_vL^∞_x(m)}, where {m˜ (1+ |v|^k)} is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an {L^2-L^∞} theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the {L^1_vL^∞_x} framework is dealt with for any {k > k_0}, recovering the optimal physical threshold of finite energy {k_0=2} in the particular case of a multi-species hard spheres mixture with the same masses.
Sels, Dries; Brosens, Fons
2013-10-01
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.
Formulating Weak Lensing from the Boltzmann Equation and Application to Lens-lens Couplings
Su, S -C
2014-01-01
The Planck mission has conclusively detected lensing of the Cosmic Microwave Background (CMB) radiation from foreground sources to an overall significance of greater than $25\\sigma$. The high precision of this measurement motivates the development of a more complete formulation of the calculation of this effect. While most effects on the CMB anisotropies are widely studied through direct solutions of the Boltzmann equation, the non-linear effect of CMB lensing is formulated through the solutions of the geodesic equation. In this paper, we present a new formalism to the calculation of the lensing effect by \\emph{directly solving the Boltzmann equation}, as we did in the calculation of the CMB anisotropies at recombination. In particular, we developed a diagrammatic approach to efficiently keep track of all the interaction terms and calculate all possible non-trivial correlations to arbitrary high orders. Using this formalism, we explicitly articulate the approximations required to recover the usual remapping a...
Coclite, Alessandro; Pascazio, Giuseppe; Decuzzi, Paolo
2016-01-01
Modelling the vascular transport and adhesion of man-made particles is crucial for optimizing their efficacy in the detection and treatment of diseases. Here, a Lattice Boltzmann and Immersed Boundary methods are combined together for predicting the near wall dynamics of particles with different shapes in a laminar flow. For the lattice Boltzmann modelling, a Gauss-Hermite projection is used to derive the lattice equation, wall boundary conditions are imposed through the Zou-He framework, and a moving least squares algorithm accurately reconstructs the forcing term accounting for the immersed boundary. First, the computational code is validated against two well-known test cases: the sedimentation of circular and elliptical cylinders in a quiescent fluid. A very good agreement is observed between the present results and those available in the literature. Then, the transport of circular, elliptical, rectangular, square and triangular particles is analyzed in a Couette flow, at Re=20. All particles drifted later...
Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers
Abramov, Rafail V
2016-01-01
In the current work we propose a diffusive modification of the Boltzmann equation. This naturally leads to the corresponding diffusive fluid dynamics equations, which we numerically investigate in a simple Couette flow setting. This diffusive modification is based on the assumption of the "imperfect" model collision term, which is unable to track all collisions in the corresponding real gas particle system. The effect of missed collisions is then modeled by an appropriately scaled long-term homogenization process of the particle dynamics. The corresponding diffusive fluid dynamics equations are produced in a standard way by closing the hierarchy of the moment equations using either the Euler or the Grad closure. In the numerical experiments with the Couette flow, we discover that the diffusive Euler equations behave similarly to the conventional Navier-Stokes equations, while the diffusive Grad equations additionally exhibit Knudsen-like velocity boundary layers. We compare the simulations with the correspond...
Global solutions in the critical Besov space for the non-cutoff Boltzmann equation
Morimoto, Yoshinori; Sakamoto, Shota
2016-10-01
The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin-Lerner space. In order to analyze the collisional term of the equation, a Chemin-Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an a priori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn-Banach extension theorem, the desired solution is obtained.
A novel protocol for linearization of the Poisson-Boltzmann equation
Tsekov, R
2014-01-01
A new protocol for linearization of the Poisson-Boltzmann equation is proposed and the resultant electrostatic equation coincides formally with the Debye-Huckel equation, the solution of which is well known for many electrostatic problems. The protocol is examined on the example of electrostatically stabilized nano-bubbles and it is shown that stable nano-bubbles could be present in aqueous solutions of anionic surfactants near the critical temperature, if the surface potential is constant. At constant surface charge non nano-bubbles could exist.
Energy Technology Data Exchange (ETDEWEB)
EL Safadi, M
2007-03-15
We study the regularity of kinetic equations of Boltzmann type.We use essentially Littlewood-Paley method from harmonic analysis, consisting mainly in working with dyadics annulus. We shall mainly concern with the homogeneous case, where the solution f(t,x,v) depends only on the time t and on the velocities v, while working with realistic and singular cross-sections (non cutoff). In the first part, we study the particular case of Maxwellian molecules. Under this hypothesis, the structure of the Boltzmann operator and his Fourier transform write in a simple form. We show a global C{sup {infinity}} regularity. Then, we deal with the case of general cross-sections with 'hard potential'. We are interested in the Landau equation which is limit equation to the Boltzmann equation, taking in account grazing collisions. We prove that any weak solution belongs to Schwartz space S. We demonstrate also a similar regularity for the case of Boltzmann equation. Let us note that our method applies directly for all dimensions, and proofs are often simpler compared to other previous ones. Finally, we finish with Boltzmann-Dirac equation. In particular, we adapt the result of regularity obtained in Alexandre, Desvillettes, Wennberg and Villani work, using the dissipation rate connected with Boltzmann-Dirac equation. (author)
Premnath, Kannan N; Banerjee, Sanjoy
2008-01-01
Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The use of multiple relaxation times in the GLBE allows enhancement of the numerical stability. Particular focus is given in preconditioning external forces, which can be spatially and temporally dependent. In particular, correct forms of moment-projections of source/forcing terms are derived such that they recover preconditioned Navier-Stokes equations with non-uniform external forces. As an illustration, we solve an extende...
Xie, Dexuan; Jiang, Yi
2016-10-01
The nonlocal dielectric approach has been studied for more than forty years but only limited to water solvent until the recent work of Xie et al. (2013) [20]. As the development of this recent work, in this paper, a nonlocal modified Poisson-Boltzmann equation (NMPBE) is proposed to incorporate nonlocal dielectric effects into the classic Poisson-Boltzmann equation (PBE) for protein in ionic solvent. The focus of this paper is to present an efficient finite element algorithm and a related software package for solving NMPBE. Numerical results are reported to validate this new software package and demonstrate its high performance for protein molecules. They also show the potential of NMPBE as a better predictor of electrostatic solvation and binding free energies than PBE.
Asinari, Pietro
2010-01-01
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both ...
Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation
Holst, Michael; Yu, Zeyun; Zhou, Yongcheng; Zhu, Yunrong
2010-01-01
We develop an efficient and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the regularization technique of Chen, Holst, and Xu; this technique made possible the first a priori pointwise estimates and the first complete solution and approximation theory for the Poisson-Boltzmann equation. It also made possible the first provably convergent discretization of the PBE, and allowed for the development of a provably convergent AFEM for the PBE. However, in practice the regularization turns out to be numerically ill-conditioned. In this article, we examine a second regularization, and establish a number of basic results to ensure that the new approach produces the same mathematical advantages of the original regularization, without the ill-conditioning property. We then design an AFEM scheme based on the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This res...
Patel, R.A.; Perko, J.; Jaques, D.; De Schutter, G.; Ye, G.; Van Breugel, K.
2013-01-01
A Lattice Boltzmann (LB) based reactive transport model intended to capture reactions and solid phase changes occurring at the pore scale is presented. The proposed approach uses LB method to compute multi component mass transport. The LB multi-component transport model is then coupled with the well
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method
Wu, Lei; White, Craig; Scanlon, Thomas J.; Reese, Jason M.; Zhang, Yonghao
2013-10-01
The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard-Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev-Krook-Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss-Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard-Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases
EXISTENCE OF INFINITE ENERGY SOLUTION TO THE INELASTIC BOLTZMANN EQUATION WITH EXTERNAL FORCE
Institute of Scientific and Technical Information of China (English)
Wei Jinbo; Zhang Xianwen
2012-01-01
In this paper,the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy.More precisely,under the assumptions on the bicharacteristic generated by external force,we prove the global existence of solution for small initial data compared to the local Maxwellian exp{-p|x-v|2},which has infinite mass and energy.
Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson-Walker space-time
Takou, Etienne
2016-01-01
In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson-Walker space-time. Unique global (in time) mild solution is obtained in a suitable weighted space.
A new exact solution of the relativistic Boltzmann equation and its hydrodynamic limit
Denicol, Gabriel S; Martinez, Mauricio; Noronha, Jorge; Strickland, Michael
2014-01-01
We present an exact solution of the relativistic Boltzmann equation for a system undergoing boost-invariant longitudinal and azimuthally symmetric transverse flow ("Gubser flow"). The resulting exact non-equilibrium dynamics is compared to 1st- and 2nd-order relativistic hydrodynamic approximations for various shear viscosity to entropy density ratios. This novel solution can be used to test the validity and accuracy of different hydrodynamic approximations in conditions similar to those generated in relativistic heavy-ion collisions.
Chatterjee, Dipankar; Amiroudine, Sakir
2011-02-01
A comprehensive non-isothermal Lattice Boltzmann (LB) algorithm is proposed in this article to simulate the thermofluidic transport phenomena encountered in a direct-current (DC) magnetohydrodynamic (MHD) micropump. Inside the pump, an electrically conducting fluid is transported through the microchannel by the action of an electromagnetic Lorentz force evolved out as a consequence of the interaction between applied electric and magnetic fields. The fluid flow and thermal characteristics of the MHD micropump depend on several factors such as the channel geometry, electromagnetic field strength and electrical property of the conducting fluid. An involved analysis is carried out following the LB technique to understand the significant influences of the aforementioned controlling parameters on the overall transport phenomena. In the LB framework, the hydrodynamics is simulated by a distribution function, which obeys a single scalar kinetic equation associated with an externally imposed electromagnetic force field. The thermal history is monitored by a separate temperature distribution function through another scalar kinetic equation incorporating the Joule heating effect. Agreement with analytical, experimental and other available numerical results is found to be quantitative.
Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation
Institute of Scientific and Technical Information of China (English)
LI; Zhihui; ZHANG; Hanxin; FU; Song
2005-01-01
The gas-kinetic unified algorithm using Boltzmann model equation have been extended and developed to solve the micro-scale gas flows in Poiseuille-like micro-channels from Micro-Electro-Mechanical Systems (MEMS). The numerical modeling of the gas kinetic boundary conditions suitable for micro-scale gas flows is presented. To test the present method, the classical Couette flows with various Knudsen numbers, the gas flows from short microchannels like plane Poiseuille and the pressure-driven gas flows in two-dimensional short microchannels have been simulated and compared with the approximate solutions of the Boltzmann equation, the related DSMC results, the modified N-S solutions with slip-flow boundary theory, the gas-kinetic BGK-Burnett solutions and the experimental data. The comparisons show that the present gas-kinetic numerical algorithm using the mesoscopic Boltzmann simplified velocity distribution function equation can effectively simulate and reveal the gas flows in microchannels. The numerical experience indicates that this method may be a powerful tool in the numerical simulation of micro-scale gas flows from MEMS.
A multi-term solution of the space-time Boltzmann equation for electrons in gaseous and liquid Argon
Boyle, G J; Tattersall, W J; McEachran, R P; White, R D
2015-01-01
In a recent paper [1] the scattering and transport of excess electrons in liquid argon in the hydrodynamic regime was investigated, generalizing the seminal works of Lekner and Cohen [2,3] with modern scattering theory techniques and kinetic theory. In this paper, the discussion is extended to the non-hydrodynamic regime through the development of a full multi-term space-time solution of Boltzmann's equation for electron transport in gases and liquids using a novel operator-splitting method. A Green's function formalism is considered that enables flexible adaptation to various experimental systems. The spatio-temporal evolution of electrons in liquids in the hydrodynamic regime is studied for a benchmark model Percus-Yevick liquid as well as for liquid argon. The temporal evolution of Franck-Hertz oscillations are observed for liquids, with striking differences in the spatio-temporal development of the velocity distribution function components between the uncorrelated gas and true liquid approximations in arg...
Solution Poisson-Boltzmann equation: Application in the Human Neuron Membrane
Soares, M A G; Cortez, C M
2008-01-01
With already demonstrated in previous work the equations that describe the space dependence of the electric potential are determined by the solution of the equation of Poisson-Boltzmann. In this work we consider these solutions for the membrane of the human neuron, using a model simplified for this structure considering the distribution of electrolytes in each side of the membrane, as well as the effect of glycocalyx and the lipidic bilayer. It was assumed that on both sides of the membrane the charges are homogeneously distributed and that the potential depends only on coordinate z.
Denicol, Gabriel S; Martinez, Mauricio; Noronha, Jorge; Strickland, Michael
2014-01-01
We present an exact solution to the Boltzmann equation which describes a system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion for arbitrary shear viscosity to entropy density ratio. This new solution is constructed by considering the conformal map between Minkowski space and the direct product of three dimensional de Sitter space with a line. The resulting solution respects SO(3)_q x SO(1,1) x Z_2 symmetry. We compare the exact kinetic solution with exact solutions of the corresponding macroscopic equations with the same symmetry that were obtained from the kinetic theory in ideal and second-order viscous hydrodynamic approximations.
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
Ayi, Nathalie
2017-01-01
We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
Ayi, Nathalie
2017-03-01
We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.
Theory of the Lattice Boltzmann Equation: Symmetry properties of Discrete Velocity Sets
Rubinstein, Robert; Luo, Li-Shi
2007-01-01
In the lattice Boltzmann equation, continuous particle velocity space is replaced by a finite dimensional discrete set. The number of linearly independent velocity moments in a lattice Boltzmann model cannot exceed the number of discrete velocities. Thus, finite dimensionality introduces linear dependencies among the moments that do not exist in the exact continuous theory. Given a discrete velocity set, it is important to know to exactly what order moments are free of these dependencies. Elementary group theory is applied to the solution of this problem. It is found that by decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group, it becomes relatively straightforward to assess the behavior of moments in the theory. The construction of some standard two- and three-dimensional models is reviewed from this viewpoint, and procedures for constructing some new higher dimensional models are suggested.
Jungemann, C.; Pham, A. T.; Meinerzhagen, B.; Ringhofer, C.; Bollhöfer, M.
2006-07-01
The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.
Perturbative and non-perturbative aspects non-Abelian Boltzmann-Langevin equation
Energy Technology Data Exchange (ETDEWEB)
Boedeker, Dietrich. E-mail: bodeker@physik.uni-bielefeld.de
2002-12-30
We study the Boltzmann-Langevin equation which describes the dynamics of hot Yang-Mills fields with typical momenta of order of the magnetic screening scale g{sup 2}T. It is transformed into a path integral and Feynman rules are obtained. We find that the leading log Langevin equation can be systematically improved in a well behaved expansion in log(1/g){sup -1}. The result by Arnold and Yaffe that the leading log Langevin equation is still valid at next-to-leading-log order is confirmed. We also confirm their result for the next-to-leading-log damping coefficient, or color conductivity, which is shown to be gauge fixing independent for a certain class of gauges. The frequency scale g{sup 2}T does not contribute to this result, but it does contribute, by power counting, to the transverse gauge field propagator. Going beyond a perturbative expansion we find 1-loop ultraviolet divergences which cannot be removed by renormalizing the parameters in the Boltzmann-Langevin equation.
Fokker-Planck Equation for Boltzmann-type and Active Particles transfer probability approach
Trigger, S A
2002-01-01
Fokker-Planck equation with the velocity-dependent coefficients is considered for various isotropic systems on the basis of probability transition (PT) approach. This method provides the self-consistent and universal description of friction and diffusion for Brownian particles. Renormalization of the friction coefficient is shown to occur for two dimensional (2-D) and three dimensional (3-D) cases, due to the tensorial character of diffusion. The specific forms of PT are calculated for the Boltzmann-type of collisions and for the absorption-type of collisions (the later are typical for dusty plasmas and some other systems). Validity of the Einstein's relation for the Boltzmann-type collisions is proved for the velocity-dependent friction and diffusion coefficients. For non-Boltzmann collisions, such as, e.g., absorption collisions, the Einstein relation is violated, although some other relations (determined by the structure of PT) can exist. The collecting part of the ion drag force in a dusty plasma, arising...
Energy Technology Data Exchange (ETDEWEB)
Ayissi, Raoul Domingo, E-mail: raoulayissi@yahoo.fr; Noutchegueme, Norbert, E-mail: nnoutch@yahoo.fr [Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon)
2015-01-15
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
A novel incompressible finite-difference lattice Boltzmann equation for particle-laden flow
Institute of Scientific and Technical Information of China (English)
Sheng Chen; Zhaohui Liu; Baochang Shi; Zhu He; Chuguang Zheng
2005-01-01
In this paper, we propose a novel incompressible finite-difference lattice Boltzmann Equation (FDLBE). Because source terms that reflect the interaction between phases can be accurately described, the new model is suitable for simulating two-way coupling incompressible multiphase flow.The 2-D particle-laden flow over a backward-facing step is chosen as a test case to validate the present method. Favorable results are obtained and the present scheme is shown to have good prospects in practical applications.
Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff
Morimoto, Yoshinori; Wang, Shuaikun; Yang, Tong
2016-12-01
A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.
Numerical solution of the Boltzmann equation for the shock wave in a gas mixture
Raines, A A
2014-01-01
We study the structure of a shock wave for a two-, three- and four-component gas mixture on the basis of numerical solution of the Boltzmann equation for the model of hard sphere molecules. For the evaluation of collision integrals we use the Conservative Projection Method developed by F.G. Tscheremissine which we extended to gas mixtures in cylindrical coordinates. The transition from the upstream to downstream uniform state is presented by macroscopic values and distribution functions. The obtained results were compared with numerical and experimental results of other authors.
Heinz, U; Denicol, G S; Martinez, M; Nopoush, M; Noronha, J; Ryblewski, R; Strickland, M
2015-01-01
Several recent results are reported from work aiming to improve the quantitative precision of relativistic viscous fluid dynamics for relativistic heavy-ion collisions. The dense matter created in such collisions expands in a highly anisotropic manner. Due to viscous effects this also renders the local momentum distribution anisotropic. Optimized hydrodynamic approaches account for these anisotropies already at leading order in a gradient expansion. Recently discovered exact solutions of the relativistic Boltzmann equation in anisotropically expanding systems provide a powerful testbed for such improved hydrodynamic approximations. We present the latest status of our quest for a formulation of relativistic viscous fluid dynamics that is optimized for applications to relativistic heavy-ion collisions.
On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates
Lu, Xuguang; Mouhot, Clément
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.
Solution of the Boltzmann Equation for Electrons in Laser-Heated Metals
Pietanza, L. D.; Colonna, G.; Capitelli, M.
2005-05-01
A kinetic study of the electron relaxation dynamic inside a noble metal film (Ag) subjected to a femtosecond laser pulse has been performed. A time dependent numerical algorithm for the solution of the Boltzmann equations for electrons and phonons inside the film has been developped, considering electron-electron and electron-phonon collisions and the laser perturbation. The dependence of electron-electron and electron-phonon characteristic time-scales on the screening parameter values has been investigated. Electron-electron relaxation times are also compared with experimental data obtained through time-resolved two-photon photoemission technique.
Leptogenesis with heavy neutrino flavours: from density matrix to Boltzmann equations
Blanchet, Steve; Di Bari, Pasquale; Marzola, Luca
2011-01-01
Leptogenesis with heavy neutrino flavours is discussed within a density matrix formalism. We write the density matrix equation that describes the generation of the matter-antimatter asymmetry, for an arbitrary choice of the right-handed (RH) neutrino masses. For hierarchical RH neutrino masses lying in the fully flavoured regimes, the density matrix equation reduces to multiple-stage Boltzmann equations. In this case we recover and extend results previously derived within a quantum state collapse description. We confirm the generic existence of phantom terms, which are not washed out at production and contribute to the flavoured asymmetries proportionally to the initial RH neutrino abundances. Even in the N_1-dominated scenario they can give rise to lepton flavour asymmetries much larger than the baryon asymmetry with potential applications. We also confirm that there is a (orthogonal) component in the asymmetry produced by the heavier RH neutrinos which completely escapes the washout from the lighter RH neut...
Wang, Huimin
2017-01-01
In this paper, a new lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher- order moments of equilibrium distribution functions are obtained. In order to make the scheme obey the three conservation laws of the KdV equation, two equilibrium distribution functions are used and a correlation between the first conservation law and the second conservation law is constructed. In numerical examples, the numerical results of the KdV equation obtained by this scheme are compared with those results obtained by the previous lattice Boltzmann model. Numerical experiments demonstrate this scheme can be used to reduce the truncation error of the lattice Boltzmann scheme and preserve the three conservation laws.
Institute of Scientific and Technical Information of China (English)
蔡新景; 王新新; 邹晓兵; 鲁志伟
2016-01-01
The propagation properties of streamer could be obtained accurately by using the fluid model to simulate, provided that precise electron transport and reaction coefficients were input the model. Presently, there were two methods to calculate the transport coefficients: interpolation method based on the data of swarm experiment or two-term approximation of the Boltzmann equation. It was showed that interpolation method could be merely used in low reduced electric condition, and for the second method the reasonableness of the isotropic assumption and the accuracy of the results were a question. In view of the above problems, multi-term approximation of the Boltzmann equation to obtain electron transport coefficients was proposed by Nesset al. Here two improvements are provided: firstly, the expansion sequence is adjusted to derive the unified hierarchy in the hydrodynamic and non-hydrodynamic limit; secondly, the collision integral is evaluated based on the Gauss-Kronrod rule instead of Gauss-Laguerre rule as used in Ness''s works. In the end hard sphere model and Reid''s ramp inelastic model are considered. It is shown that it is more accurate to evaluate the collision integral based on Gauss-Kronrod integration method than Gauss-Laguerre method. Furthermore, it is demonstrated that the electron velocity distribution is anisotropic even only undergoing conservative collision, so there is a great error if the two -term approximation is used to obtain the electron transport coefficients.%获得准确的电子输运和反应系数是采用流体模型准确仿真流注传播特性的前提.目前,电子输运系数主要有两种计算方法:一是用电子群实验数据进行插值;二是用两项近似方法解玻尔兹曼方程.方法一只能用在约化场强很小的场合,方法二的理论基础即各相同性假设是否成立和数据准确度尚无定论.针对以上问题,Ness 等人采用了多项近似法解玻尔兹曼方程计算电子输运系数.该
Düring, Bertram
2015-01-01
We propose and investigate different kinetic models for opinion formation, when the opinion formation process depends on an additional independent variable, e.g. a leadership or a spatial variable. More specifically, we consider:(i) opinion dynamics under the effect of opinion leadership, where each individual is characterised not only by its opinion, but also by another independent variable which quantifies leadership qualities; (ii) opinion dynamics modelling political segregation in the `The Big Sort', a phenomenon that US citizens increasingly prefer to live in neighbourhoods with politically like-minded individuals. Based on microscopic opinion consensus dynamics such models lead to inhomogeneous Boltzmann-type equations for the opinion distribution. We derive macroscopic Fokker-Planck-type equations in a quasi-invariant opinion limit and present results of numerical experiments.
AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.
Koehl, Patrice; Delarue, Marc
2010-02-14
The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE
Lattice Boltzmann simulations for proton transport in 2-D model channels of Nafion.
Akinaga, Yoshinobu; Hyodo, Shi-aki; Ikeshoji, Tamio
2008-10-01
Proton conductance in a 2-D channel with a slab-like structure was studied to verify that the lattice Boltzmann method (LBM) can be used as a simulation tool for proton conduction in a Nafion membrane, which is a mesoscopic system with a highly disordered porous structure. Diffusion resulting from a concentration gradient and migration by an electrostatic force were considered as the origins of proton transport. The electrostatic force acting on a proton was computed by solving the Poisson equation. The proton concentration in the membrane is expressed as a continuous function and the sulfonic charge is placed discretely. The space-averaged conductance of protons in a nonequilibrium stationary state was evaluated as a function of the structural parameters: namely, channel width and distribution of the sulfonic groups. The resulting space-averaged conductance deviates from the bulk values, depending particularly on the sulfonic group distribution. Details of the simulation scheme are described and the applicability of the present scheme to real membranes is discussed.
Institute of Scientific and Technical Information of China (English)
YUAN Hongjun; YAN Han
2009-01-01
The existence and uniqueness of the solutions for the Boltzmann equations with measures as initial value are still an open problem which is posed by P. L. Lions (2000). The aim of this paper is to discuss the Cauchy problem of the system of discrete Boltzmann equations of the form etf∫i+(∫mii)x=Qi(∫1,∫2,…,∫n), (mi1, i=1,…,n) with non-negative finite Radon measures as initial conditions. In particular, the existence and uniqueness of BV solutions for the above problem are obtained.
Directory of Open Access Journals (Sweden)
Khoie R.
1996-01-01
Full Text Available A self-consistent Boltzmann-Poisson-Schrödinger solver for High Electron Mobility Transistor is presented. The quantization of electrons in the quantum well normal to the heterojunction is taken into account by solving the two higher moments of Boltzmann equation along with the Schrödinger and Poisson equations, self-consistently. The Boltzmann transport equation in the form of a current continuity equation and an energy balance equation are solved to obtain the transient and steady-state transport behavior. The numerical instability problems associated with the simulator are presented, and the criteria for smooth convergence of the solutions are discussed. The current-voltage characteristics, transconductance, gate capacitance, and unity-gain frequency of a single quantum well HEMT is discussed. It has been found that a HEMT device with a gate length of 0.7 μ m , and with a gate bias voltage of 0.625 V, has a transconductance of 579.2 mS/mm, which together with the gate capacitance of 19.28 pF/cm, can operate at a maximum unity-gain frequency of 47.8 GHz.
Directory of Open Access Journals (Sweden)
R. Khoie
1996-01-01
Full Text Available A self-consistent Boltzmann-Poisson-Schrödinger solver for High Electron Mobility Transistor is presented. The quantization of electrons in the quantum well normal to the heterojunction is taken into account by solving the two higher moments of Boltzmann equation along with the Schrödinger and Poisson equations, self-consistently. The Boltzmann transport equation in the form of a current continuity equation and an energy balance equation are solved to obtain the transient and steady-state transport behavior. The numerical instability problems associated with the simulator are presented, and the criteria for smooth convergence of the solutions are discussed. The current-voltage characteristics, transconductance, gate capacitance, and unity-gain frequency of a single quantum well HEMT is discussed. It has been found that a HEMT device with a gate length of 0.7 μm, and with a gate bias voltage of 0.625 V, has a transconductance of 579.2 mS/mm, which together with the gate capacitance of 19.28 pF/cm, can operate at a maximum unity-gain frequency of 47.8 GHz.
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
Directory of Open Access Journals (Sweden)
José Colmenares
2014-01-01
Full Text Available The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.
A combined MPI-CUDA parallel solution of linear and nonlinear Poisson-Boltzmann equation.
Colmenares, José; Galizia, Antonella; Ortiz, Jesús; Clematis, Andrea; Rocchia, Walter
2014-01-01
The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.
Arnab, Sarkar; Manjeet, Singh
2017-02-01
We report spectroscopic studies on plasma electron number density of laser-induced plasma produced by ns-Nd:YAG laser light pulses on an aluminum sample in air at atmospheric pressure. The effect of different laser energy and the effect of different laser wavelengths were compared. The experimentally observed line profiles of neutral aluminum have been used to extract the excitation temperature using the Boltzmann plot method, whereas the electron number density has been determined from the Stark broadened as well as using the Saha-Boltzmann equation (SBE). Each approach was also carried out by using the Al emission line and Mg emission lines. It was observed that the SBE method generated a little higher electron number density value than the Stark broadening method, but within the experimental uncertainty range. Comparisons of N e determined by the two methods show the presence of a linear relation which is independent of laser energy or laser wavelength. These results show the applicability of the SBE method for N e determination, especially when the system does not have any pure emission lines whose electron impact factor is known. Also use of Mg lines gives superior results than Al lines.
Onsager's-principle-consistent 13-moment transport equations.
Singh, Narendra; Agrawal, Amit
2016-06-01
A new set of generalized transport equations is derived for higher-order moments which are generated in evolution equation for stress tensor and heat flux vector in 13-moment equations. The closure we employ satisfies Onsager's symmetry principle. In the derivation, we do not employ a phase density function based on Hermite polynomial series in terms of higher-order moments, unlike Grad's approach. The distribution function is rather chosen to satisfy collision invariance, and H-theorem and capture relatively strong deviations from equilibrium. The phase density function satisfies the linearized Boltzmann equation and provides the correct value of the Prandtl number for monatomic gas. The derived equations are compared with Grad's 13-moments equations for gas modeled as Maxwellian molecule. The merits of the proposed equations against Grad's and R13 equations are discussed. In particular, it is noted that the proposed equations contain higher-order terms compared to these equations but require a fewer number of boundary conditions as compared to the R13 equations. The Knudsen number envelope which can be covered to describe flows with these equations is therefore expected to be larger as compared to the earlier equations.
LATTICE BOLTZMANN METHOD SIMULATIONS FOR MULTIPHASE FLUIDS WITH REDICH-KWONG EQUATION OF STATE
Institute of Scientific and Technical Information of China (English)
WEI Yi-kun; QIAN Yue-hong
2011-01-01
In this article we state that the compression factor of the Redlich-Kwong Equation Of State (EOS) is smaller than that of van der Waals EOS.The Redlich-Kwong EOS is in better agreement with experimental data on coexistence curves at the critical point than the van der Waals EOS.We implement the Redlich-Kwong EOS in the Lattice Boltzmann Method (LBM) simulations via a pseudo-potential approach.We propose a new force,which can obtain computational stationary and reach larger density ratio.As a result,multi-phase flows with large density ratio (up to 1012 in the stationary case) can be simulated.We perform four numerical simulations,which are respectively related to single liquid droplet,vapor-liquid separation,surface tension and liquid coalescence of two droplets.
Exponential Runge-Kutta schemes for inhomogeneous Boltzmann equations with high order of accuracy
Li, Qin
2012-01-01
We consider the development of exponential methods for the robust time discretization of space inhomogeneous Boltzmann equations in stiff regimes. Compared to the space homogeneous case, or more in general to the case of splitting based methods, studied in Dimarco Pareschi (SIAM J. Num. Anal. 2011) a major difficulty is that the local Maxwellian equilibrium state is not constant in a time step and thus needs a proper numerical treatment. We show how to derive asymptotic preserving (AP) schemes of arbitrary order and in particular using the Shu-Osher representation of Runge-Kutta methods we explore the monotonicity properties of such schemes, like strong stability preserving (SSP) and positivity preserving. Several numerical results confirm our analysis.
Institute of Scientific and Technical Information of China (English)
Du Gang; Liu Xiao-Yan; Han Ru-Qi
2006-01-01
A two-dimensional (2D) full band self-consistent ensemble Monte Carlo (MC) method for solving the quantum Boltzmann equation, including collision broadening and quantum potential corrections, is developed to extend the MC method to the study of nano-scale semiconductor devices with obvious quantum mechanical (QM) effects. The quantum effects both in real space and momentum space in nano-scale semiconductor devices can be simulated. The effective mobility in the inversion layer of n and p channel MOSFET is simulated and compared with experimental data to verify this method. With this method 50nm ultra thin body silicon on insulator MOSFET are simulated. Results indicate that this method can be used to simulate the 2D QM effects in semiconductor devices including tunnelling effect.
Mirzadeh, Mohammad; Squires, Todd; Gibou, Frederic
2010-11-01
We present a finite difference discretization of the non-linear Poisson-Boltzmann (PB) equation over complex geometries that has second order accurracy. The level-set method is adopted to represent the interface and Octree (in three dimensions) or Quadtree (in two dimensions) data stuructures are used to generate adaptive grids. Such an approach garanties that the finest grid resolution is located near the interface where EDL forms and creates very large electric field. Several numerical experiments are carried which indicate the second order accuracy both in the case of Dirichlet and Neumann boundary conditions in L2 and L∞ norms. Finally, we use our method to study the electrostatic interaction of double layers between charged particles in an unbounded bulk electrolyte as well as in a channel where the channel width is of the order of Debye length.
Volume transport and generalized hydrodynamic equations for monatomic fluids.
Eu, Byung Chan
2008-10-01
In this paper, the effects of volume transport on the generalized hydrodynamic equations for a pure simple fluid are examined from the standpoint of statistical mechanics and, in particular, kinetic theory of fluids. First, we derive the generalized hydrodynamic equations, namely, the constitutive equations for the stress tensor and heat flux for a single-component monatomic fluid, from the generalized Boltzmann equation in the presence of volume transport. Then their linear steady-state solutions are derived and examined with regard to the effects of volume transport on them. The generalized hydrodynamic equations and linear constitutive relations obtained for nonconserved variables make it possible to assess Brenner's proposition [Physica A 349, 11 (2005); Physica A 349, 60 (2005)] for volume transport and attendant mass and volume velocities as well as the effects of volume transport on the Newtonian law of viscosity, compression/dilatation (bulk viscosity) phenomena, and Fourier's law of heat conduction. On the basis of study made, it is concluded that the notion of volume transport is sufficiently significant to retain in irreversible thermodynamics of fluids and fluid mechanics.
Diffusion equation and spin drag in spin-polarized transport
DEFF Research Database (Denmark)
Flensberg, Karsten; Jensen, Thomas Stibius; Mortensen, Asger
2001-01-01
We study the role of electron-electron interactions for spin-polarized transport using the Boltzmann equation, and derive a set of coupled transport equations. For spin-polarized transport the electron-electron interactions are important, because they tend to equilibrate the momentum of the two-spin...... species. This "spin drag" effect enhances the resistivity of the system. The enhancement is stronger the lower the dimension is, and should be measurable in, for example, a two-dimensional electron gas with ferromagnetic contacts. We also include spin-flip scattering, which has two effects......: it equilibrates the spin density imbalance and, provided it has a non-s-wave component, also a current imbalance....
Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations
Fan, W C; Powell, J L
2002-01-01
The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations.
Lattice Boltzmann Simulation of Multiphase Transport in Nanostructured PEM Fuel Cells
Stiles, Christopher D.
As the fossil fuel crisis becomes more critical, it is imperative to develop renewable sources of power generation. Polymer electrolyte membrane (PEM) fuel cells are considered a viable option. However, the cost of the platinum catalyst has hindered their commercialization. PEM fuel cells with platinum loading of >0.4 mg cm2 are common. Efforts towards further reducing this loading are currently underway utilizing nanostructured electrodes. A consequence of increased platinum utilization per unit area and thinner nanostructured electrodes is flooding, which is detrimental to fuel cell performance. Flooding causes a two-fold impact on cell performance: a drop in cell voltage and a rise in parasitic pumping power to overcome the increased pressure drop, which together result in a significant reduction in system efficiency. Proper water management is therefore crucial for optimum performance of the fuel cell and also for enhancing membrane durability. The goal of this thesis is to simulate the multiphase fluid transport in the nanostructured PEMFC of H2O in air with realistic density ratios. In order to pursue this goal, the ability of the pseudopotential based multiphase lattice Boltzmann method to realistically model the coexistence of the gas and liquid phases of H2O at low temperatures is explored. This method is expanded to include a gas mixture of O2 and N 2 into the multiphase H2O systems. Beginning with the examination of the phase transition region described by the current implementation of the multiphase pseudopotential lattice Boltzmann model. Following this, a modified form of the pressure term with the use of a scalar multiplier kappa for the Peng-Robinson equation of state is thoroughly investigated. This method proves to be very effective at enabling numerically stable simulations at low temperatures with large density ratios. It is found that for decreasing values of kappa, this model leads to an increase in multiphase interface thickness and a
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution).
Bazow, D.; Denicol, G. S.; Heinz, U.; Martinez, M.; Noronha, J.
2016-12-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of nonhydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the nonhydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. However, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in the relaxation time approximation, followed, in turn, by kinetic evolution with a linearized and a fully nonlinear Boltzmann collision term.
Noutchegueme, N; Noutchegueme, Norbert; Tetsadjio, Mesmin Erick
2003-01-01
We prove, for the relativistic Boltzmann equation in the homogeneous case, on the Minkowski space-time, a global in time existence and uniqueness theorem. The method we develop extends to the cases of some curved space-times such as the flat Robertson-Walker space-time and some Bianchi type I space-times.
Langevin Poisson-Boltzmann equation: point-like ions and water dipoles near a charged surface.
Gongadze, Ekaterina; van Rienen, Ursula; Kralj-Iglič, Veronika; Iglič, Aleš
2011-06-01
Water ordering near a charged membrane surface is important for many biological processes such as binding of ligands to a membrane or transport of ions across it. In this work, the mean-field Poisson-Boltzmann theory for point-like ions, describing an electrolyte solution in contact with a planar charged surface, is modified by including the orientational ordering of water. Water molecules are considered as Langevin dipoles, while the number density of water is assumed to be constant everywhere in the electrolyte solution. It is shown that the dielectric permittivity of an electrolyte close to a charged surface is decreased due to the increased orientational ordering of water dipoles. The dielectric permittivity close to the charged surface is additionally decreased due to the finite size of ions and dipoles.
Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation
Ren, Feng; Song, Baowei; Sukop, Michael C.; Hu, Haibao
2016-08-01
The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015), 10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev. E 89, 053320 (2014), 10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling.
Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation.
Fraenkel, Dan
2015-12-01
The linearized Poisson-Boltzmann (L-PB) equation is examined for its κ-range of validity (κ, Debye reciprocal length). This is done for the Debye-Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010, 108, 1435), which extends the DH theory to the case of ion-size dissimilarity (therefore dubbed DH-SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH-SiS fits with data of the mean ionic activity coefficient, γ± (molal), against m, even at m > 1 (κ > 0.33 Å(-1) ). The SiS expressions combine the overall extra-electrostatic potential energy of the smaller ion, as central ion-Ψa>b (κ), with that of the larger ion, as central ion-Ψb>a (κ); a and b are, respectively, the counterion and co-ion distances of closest approach. Ψa>b and Ψb>a are derived from the L-PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L-PB equation can be valid up to κ ≥ 1.3 Å(-1) if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean-field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L-PB equation; the lethal approximation is assigning a single size to the positive and negative ions.
Robson, R E; Winkler, R; Sigeneger, F
2002-05-01
The Boltzmann equation corresponding to a general "multiterm" representation of the phase space distribution function f(r,c,t) for charged particles in a gas in an electric field was reformulated entirely in terms of spherical tensors f(l)(m) some time ago, and numerous applications, including extension to time varying and crossed electric and magnetic fields, have followed. However, these applications have, by and large, been limited to the hydrodynamic conditions that prevail in swarm experiments and the full potential of the tensor formalism has thus never been realized. This paper resumes the discussion in the context of the more general nonhydrodynamic situation. Geometries for which a simple Legendre polynomial expansion suffices to represent f are discussed briefly, but the emphasis is upon cylindrical geometry, where such simplification does not arise. In particular, we consider an axisymmetric cylindrical column of weakly ionized plasma, and derive an infinite hierarchy of integrodifferential equations for the expansion coefficients of the phase space distribution function, valid for both electrons and ions, and for all types of binary interaction with neutral gas molecules.
Boltzmann-equation simulations of radio-frequency-driven, low-temperature plasmas
Energy Technology Data Exchange (ETDEWEB)
Drallos, P.J.; Riley, M.E.
1995-01-01
We present a method for the numerical solution of the Boltzmann equation (BE) describing plasma electrons. We apply the method to a capacitively-coupled, radio-frequency-driven He discharge in parallel-plate (quasi-1D) geometry which contains time scales for physical processes spanning six orders of magnitude. Our BE solution procedure uses the method of characteristics for the Vlasov operator with interpolation in phase space at early time, allowing storage of the distribution function on a fixed phase-space grid. By alternating this BE method with a fluid description of the electrons, or with a novel time-cycle-average equation method, we compute the periodic steady state of a He plasma by time evolution from startup conditions. We find that the results compare favorably with measured current-voltage, plasma density, and ``cited state densities in the ``GEC`` Reference Cell. Our atomic He model includes five levels (some are summed composites), 15 electronic transitions, radiation trapping, and metastable-metastable collisions.
Khajepor, Sorush; Chen, Baixin
2016-01-01
A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)]10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.
Ahrens, Cory D
2014-01-01
The classical $S_n$ equations of Carlson and Lee have been a mainstay in multi-dimensional radiation transport calculations. In this paper, an alternative to the $S_n$ equations, the "Lagrange Discrete Ordinate" (LDO) equations are derived. These equations are based on an interpolatory framework for functions on the unit sphere in three dimensions. While the LDO equations retain the formal structure of the classical $S_n$ equations, they have a number of important differences. The LDO equations naturally allow the angular flux to be evaluated in directions other than those found in the quadrature set. To calculate the scattering source in the LDO equations, no spherical harmonic moments are needed--only values of the angular flux. Moreover, the LDO scattering source preserves the eigenstructure of the continuous scattering operator. The formal similarity of the LDO equations with the $S_n$ equations should allow easy modification of mature 3D $S_n$ codes such as PARTISN or PENTRAN to solve the LDO equations. ...
Energy Technology Data Exchange (ETDEWEB)
Siewert, C.E. [North Carolina State Univ., Dept. Mathematics, Raleigh, NC (United States)
2002-10-01
A synthetic-kernel model (CES model) of the linearized Boltzmann equation is used along with an analytical discrete-ordinates method (ADO) to solve three fundamental problems concerning flow of a rarefied gas in a plane channel. More specifically, the problems of Couette flow, Poiseuille flow and thermal-creep flow are solved in terms of the CES model equation for an arbitrary mixture of specular and diffuse reflection at the walls confining the flow, and numerical results for the basic quantities of interest are reported. The comparisons made with results derived from solutions based on computationally intensive methods applied to the linearized Boltzmann equation are used to conclude that the CES model can be employed with confidence to improve the accuracy of results available from simpler approximations such as the BGK model or the S model. (author)
Energy Technology Data Exchange (ETDEWEB)
Borges, P. D., E-mail: pdborges@gmail.com, E-mail: lscolfaro@txstate.edu; Scolfaro, L., E-mail: pdborges@gmail.com, E-mail: lscolfaro@txstate.edu [Department of Physics, Texas State University, San Marcos, Texas 78666 (United States)
2014-12-14
The thermoelectric properties of indium nitride in the most stable wurtzite phase (w-InN) as a function of electron and hole concentrations and temperature were studied by solving the semiclassical Boltzmann transport equations in conjunction with ab initio electronic structure calculations, within Density Functional Theory. Based on maximally localized Wannier function basis set and the ab initio band energies, results for the Seebeck coefficient are presented and compared with available experimental data for n-type as well as p-type systems. Also, theoretical results for electric conductivity and power factor are presented. Most cases showed good agreement between the calculated properties and experimental data for w-InN unintentionally and p-type doped with magnesium. Our predictions for temperature and concentration dependences of electrical conductivity and power factor revealed a promising use of InN for intermediate and high temperature thermoelectric applications. The rigid band approach and constant scattering time approximation were utilized in the calculations.
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions
Hu, Zexi; Huang, Juntao; Yong, Wen-An
2016-04-01
In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015), 10.1016/j.jcp.2015.07.045] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.
On anisotropy function in crystal growth simulations using Lattice Boltzmann equation
Younsi, Amina
2016-01-01
In this paper, we present the ability of the Lattice Boltzmann (LB) equation, usually applied to simulate fluid flows, to simulate various shapes of crystals. Crystal growth is modeled with a phase-field model for a pure substance, numerically solved with a LB method in 2D and 3D. This study focuses on the anisotropy function that is responsible for the anisotropic surface tension between the solid phase and the liquid phase. The anisotropy function involves the unit normal vectors of the interface, defined by gradients of phase-field. Those gradients have to be consistent with the underlying lattice of the LB method in order to avoid unwanted effects of numerical anisotropy. Isotropy of the solution is obtained when the directional derivatives method, specific for each lattice, is applied for computing the gradient terms. With the central finite differences method, the phase-field does not match with its rotation and the solution is not any more isotropic. Next, the method is applied to simulate simultaneous...
Kernels of the linear Boltzmann equation for spherical particles and rough hard sphere particles.
Khurana, Saheba; Thachuk, Mark
2013-10-28
Kernels for the collision integral of the linear Boltzmann equation are presented for several cases. First, a rigorous and complete derivation of the velocity kernel for spherical particles is given, along with reductions to the smooth, rigid sphere case. This combines and extends various derivations for this kernel which have appeared previously in the literature. In addition, the analogous kernel is derived for the rough hard sphere model, for which a dependence upon both velocity and angular velocity is required. This model can account for exchange between translational and rotational degrees of freedom. Finally, an approximation to the exact rough hard sphere kernel is presented which averages over the rotational degrees of freedom in the system. This results in a kernel depending only upon velocities which retains a memory of the exchange with rotational states. This kernel tends towards the smooth hard sphere kernel in the limit when translational-rotational energy exchange is attenuated. Comparisons are made between the smooth and approximate rough hard sphere kernels, including their dependence upon velocity and their eigenvalues.
Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases
Zheng, Lin; Lee, Taehun; Guo, Zhaoli; Rumschitzki, David
2014-03-01
One characteristic of multiphase lattice Boltzmann equation (LBE) methods is that the interfacial region has a finite (i.e., noninfinitesimal) thickness known as a diffuse interface. In simulations of, e.g., bubble or drop dynamics, for problems involving nonideal gases, one frequently observes that the diffuse interface method produces a spontaneous, nonphysical shrinkage of the bubble or drop radius. In this paper, we analyze in detail a single-fluid two-phase model and use a LBE model for nonideal gases in order to explain this fundamental problem. For simplicity, we only investigate the static bubble or droplet problem. We find that the method indeed produces a density shift, bubble or droplet shrinkage, as well as a critical radius below which the bubble or droplet eventually vanishes. Assuming that the ratio between the interface thickness D and the initial bubble or droplet radius r0 is small, we analytically show the existence of this density shift, bubble or droplet radius shrinkage, and critical bubble or droplet survival radius. Numerical results confirm our analysis. We also consider droplets on a solid surface with different curvatures, contact angles, and initial droplet volumes. Numerical results show that the curvature, contact angle, and the initial droplet volume have an effect on this spontaneous shrinkage process, consistent with the survival criterion.
Energy Technology Data Exchange (ETDEWEB)
Chang, B
2004-03-22
This paper contains three analytical solutions of transport problems which can be used to test ray-effect errors in the numerical solutions of the Boltzmann Transport Equation (BTE). We derived the first two solutions and the third was shown to us by M. Prasad. Since this paper is intended to be an internal LLNL report, no attempt was made to find the original derivations of the solutions in the literature in order to cite the authors for their work.
Li, Zhihui; Wu, Junlin; Ma, Qiang; Jiang, Xinyu; Zhang, Hanxin
2014-12-01
Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinate points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body.
Energy Technology Data Exchange (ETDEWEB)
Li, Zhihui; Ma, Qiang [Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, P.O.Box 211, Mianyang 621000, China and National Laboratory for Computational Fluid Dynamics, No.37 Xueyuan Road, Beijing 100191 (China); Wu, Junlin; Jiang, Xinyu [Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, P.O.Box 211, Mianyang 621000 (China); Zhang, Hanxin [National Laboratory for Computational Fluid Dynamics, No.37 Xueyuan Road, Beijing 100191 (China)
2014-12-09
Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinate points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body.
Zhang, Jie; Liu, Xiaolin; Wen, Yanwei; Shi, Lu; Chen, Rong; Liu, Huijun; Shan, Bin
2017-01-25
Good electronic transport capacity and low lattice thermal conductivity are beneficial for thermoelectric applications. In this study, the potential use as a thermoelectric material for the recently synthesized two-dimensional TiS3 monolayer is explored by applying first-principles method combined with Boltzmann transport theory. Our work demonstrates that carrier transport in the TiS3 sheet is orientation-dependent, caused by the difference in charge density distribution at band edges. Due to a variety of Ti-S bonds with longer lengths, we find that the TiS3 monolayer shows thermal conductivity much lower compared with that of transition-metal dichalcogenides such as MoS2. Combined with a high power factor along the y-direction, a considerable n-type ZT value (3.1) can be achieved at moderate carrier concentration, suggesting that the TiS3 monolayer is a good candidate for thermoelectric applications.
Chakavorty, Arghya; Li, Lin; Alexov, Emil
2016-10-30
Macromolecular interactions are essential for understanding numerous biological processes and are typically characterized by the binding free energy. Important component of the binding free energy is the electrostatics, which is frequently modeled via the solutions of the Poisson-Boltzmann Equations (PBE). However, numerous works have shown that the electrostatic component (ΔΔGelec ) of binding free energy is very sensitive to the parameters used and modeling protocol. This prompted some researchers to question the robustness of PBE in predicting ΔΔGelec . We argue that the sensitivity of the absolute ΔΔGelec calculated with PBE using different input parameters and definitions does not indicate PBE deficiency, rather this is what should be expected. We show how the apparent sensitivity should be interpreted in terms of the underlying changes in several numerous and physical parameters. We demonstrate that PBE approach is robust within each considered force field (CHARMM-27, AMBER-94, and OPLS-AA) once the corresponding structures are energy minimized. This observation holds despite of using two different molecular surface definitions, pointing again that PBE delivers consistent results within particular force field. The fact that PBE delivered ΔΔGelec values may differ if calculated with different modeling protocols is not a deficiency of PBE, but natural results of the differences of the force field parameters and potential functions for energy minimization. In addition, while the absolute ΔΔGelec values calculated with different force field differ, their ordering remains practically the same allowing for consistent ranking despite of the force field used. © 2016 Wiley Periodicals, Inc.
Gray free-energy multiphase lattice Boltzmann model with effective transport and wetting properties
Zalzale, Mohamad; Ramaioli, M.; Scrivener, K. L.; McDonald, P. J.
2016-11-01
The paper shows that it is possible to combine the free-energy lattice Boltzmann approach to multiphase modeling of fluids involving both liquid and vapor with the partial bounce back lattice Boltzmann approach to modeling effective media. Effective media models are designed to mimic the properties of porous materials with porosity much finer than the scale of the simulation lattice. In the partial bounce-back approach, an effective media parameter or bounce-back fraction controls fluid transport. In the combined model, a wetting potential is additionally introduced that controls the wetting properties of the fluid with respect to interfaces between free space (white nodes), effective media (gray nodes), and solids (black nodes). The use of the wetting potential combined with the bounce-back parameter gives the model the ability to simulate transport and sorption of a wide range of fluid in material systems. Results for phase separation, permeability, contact angle, and wicking in gray media are shown. Sorption is explored in small sections of model multiscale porous systems to demonstrate two-step desorption, sorption hysteresis, and the ink-bottle effect.
Lattice Boltzmann Model for the Coupled Korteweg-de Vries Equations%耦合KdV方程组的格子Boltzmann模型
Institute of Scientific and Technical Information of China (English)
王慧敏; 刘艳红
2012-01-01
用格子Boltzmann方法研究耦合KdV方程组.构建耦合KdV方程组的格子Boltzmann模型并进行了数值实验,同时将格子Boltzmann解与其他传统数值方法得到的数值解进行比较.结果表明,格子Boltzmann方法是一种求解耦合KdV方程组的有效方法.%We used the lattice Boltzmann method to study the coupled Korteweg-de Vries equations. We constructed a lattice Boltzmann model for the coupled Korteweg-de Vries equations, and performed numerical experiments. Comparing the lattice Boltzmann solution with other numerical solutions, we obtained that the lattice Boltzmann method is an effective method for simulating the coupled Korteweg-de Vries equations.
S, Savithiri; Pattamatta, Arvind; Das, Sarit K
2015-01-01
Severe contradictions exist between experimental observations and computational predictions regarding natural convective thermal transport in nanosuspensions. The approach treating nanosuspensions as homogeneous fluids in computations has been pin pointed as the major contributor to such contradictions. To fill the void, inter particle and particle fluid interactivities (slip mechanisms), in addition to effective thermophysical properties, have been incorporated within the present formulation. Through thorough scaling analysis, the dominant slip mechanisms have been identified. A Multi Component Lattice Boltzmann Model (MCLBM) approach has been proposed, wherein the suspension has been treated as a non homogeneous twin component mixture with the governing slip mechanisms incorporated. The computations based on the mathematical model can accurately predict and quantify natural convection thermal transport in nanosuspensions. The role of slip mechanisms such as Brownian diffusion, thermophoresis, drag, Saffman ...
Lattice Boltzmann equation calculation of internal, pressure-driven turbulent flow
Hammond, L A; Care, C M; Stevens, A
2002-01-01
We describe a mixing-length extension of the lattice Boltzmann approach to the simulation of an incompressible liquid in turbulent flow. The method uses a simple, adaptable, closure algorithm to bound the lattice Boltzmann fluid incorporating a law-of-the-wall. The test application, of an internal, pressure-driven and smooth duct flow, recovers correct velocity profiles for Reynolds number to 1.25 x 10 sup 5. In addition, the Reynolds number dependence of the friction factor in the smooth-wall branch of the Moody chart is correctly recovered. The method promises a straightforward extension to other curves of the Moody chart and to cylindrical pipe flow.
Conjugate heat and mass transfer in the lattice Boltzmann equation method.
Li, Like; Chen, Chen; Mei, Renwei; Klausner, James F
2014-04-01
An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, (iii) steady heat conduction inside a circular domain with two different solid materials, and (iv) unsteady mass transfer from a spherical droplet in an extensional creeping flow. The accuracy and order of convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration), and heat (mass) flux are examined in detail and compared with those obtained from the "half-lattice division" treatment in the literature. The present analysis and numerical results show that the half-lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links, while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved
Chen, Li; Kang, Qinjun; Yao, Jun; Tao, Wenquan
2014-01-01
Porous structures of shales are reconstructed based on scanning electron microscopy (SEM) images of shale samples from Sichuan Basin, China. Characterization analyzes of the nanoscale reconstructed shales are performed, including porosity, pore size distribution, specific surface area and pore connectivity. The multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) fluid flow model and single-relaxation-time (SRT) LBM diffusion model are adopted to simulate the fluid flow and Knudsen diffusion process within the reconstructed shales, respectively. Tortuosity, intrinsic permeability and effective Knudsen diffusivity are numerically predicted. The tortuosity is much higher than that commonly employed in Bruggeman equation. Correction of the intrinsic permeability by taking into consideration the contribution of Knudsen diffusion, which leads to the apparent permeability, is performed. The correction factor under different Knudsen number and pressure are estimated and compared with existing corrections re...
A Discussion on Whether 15-20C Are All Skin Nuclei via Isospin-dependent Boltzmann-Langevin Equation
Institute of Scientific and Technical Information of China (English)
CHEN Yu; ZHANG Feng-Shou; SU Jun
2009-01-01
A new attempt of calculation for the total reaction cross sections (σR) has been carried out within the isospin-dependent Boltzmann-Langevin equation in the intermediate energy heavy-ion collision of isotopes of G. The σR of both stable and exotic nuclei are reproduced rather well. The incident energy and isospin dependencies of σR have been investigated. It is found that the isospin effect is comparatively remarkable at intermediate energy. It is also found that ~(15-18)C are neutron skin nuclei but for ~(19)C and ~(20)C we cannot draw a conclusion whether they have halo structures.
Lattice Boltzmann technique for heat transport phenomena coupled with melting process
Ibrahem, A. M.; El-Amin, M. F.; Mohammadein, A. A.; Gorla, Rama Subba Reddy
2017-01-01
In this work, the heat transport phenomena coupled with melting process are studied by using the enthalpy-based lattice Boltzmann method (LBM). The proposed model is a modified version of thermal LB model, where could avoid iteration steps and ensures high accuracy. The Bhatnagar-Gross-Krook (BGK) approximation with a D1Q2 lattice was used to determine the temperature field for one-dimensional melting by conduction and multi-distribution functions (MDF) with D2Q9 lattice was used to determine the density, velocity and temperature fields for two-dimensional melting by natural convection. Different boundary conditions including Dirichlet, adiabatic and bounce-back boundary conditions were used. The influence of increasing Rayleigh number (from 103 to 105) on temperature distribution and melting process is studied. The obtained results show that a good agreement with the analytical solution for melting by conduction case and with the benchmark solution for melting by convection.
Lattice Boltzmann technique for heat transport phenomena coupled with melting process
Ibrahem, A. M.; El-Amin, M. F.; Mohammadein, A. A.; Gorla, Rama Subba Reddy
2016-04-01
In this work, the heat transport phenomena coupled with melting process are studied by using the enthalpy-based lattice Boltzmann method (LBM). The proposed model is a modified version of thermal LB model, where could avoid iteration steps and ensures high accuracy. The Bhatnagar-Gross-Krook (BGK) approximation with a D1Q2 lattice was used to determine the temperature field for one-dimensional melting by conduction and multi-distribution functions (MDF) with D2Q9 lattice was used to determine the density, velocity and temperature fields for two-dimensional melting by natural convection. Different boundary conditions including Dirichlet, adiabatic and bounce-back boundary conditions were used. The influence of increasing Rayleigh number (from 103 to 105) on temperature distribution and melting process is studied. The obtained results show that a good agreement with the analytical solution for melting by conduction case and with the benchmark solution for melting by convection.
Lattice Boltzmann Method for Diffusion-Reaction-Transport Processes in Heterogeneous Porous Media
Institute of Scientific and Technical Information of China (English)
XU You-Sheng; ZHONG Yi-Jun; HUANG Guo-Xiang
2004-01-01
Based on the lattice Boltzmann method and general theory of fluids flowing in porous media, a numerical model is presented for the diffusion-reaction-transport (DRT) processes in porous media. As a test, we simulate a DRT process in a two-dimensional horizontal heterogeneous porous medium. The influence of gravitation in this case can be neglected, and the DRT process can be described by a strongly heterogeneous diagnostic test strip or a thin confined piece of soil with stochastically distributing property in horizontal directions. The results obtained for the relations between reduced fluid saturation S, concentration c1, and concentration c2 are shown by using the visualization computing technique. The computational efficiency and stability of the model are satisfactory.
Mixing and Transport in the Small Intestine: A Lattice-Boltzmann Model
Banco, Gino; Brasseur, James; Wang, Yanxing; Aliani, Amit; Webb, Andrew
2007-11-01
The two primary functions of the small intestine are absorption of nutrients into the blood stream and transport of material along the gut for eventual evacuation. The primary transport mechanism is peristalsis. The time scales for absorption, however, rely on mixing and transport of molecules between the bulk flow and epithelial surface. Two basic motions contribute to mixing: peristalsis and repetitive segmental contraction of short segments of the gut. In this study we evaluate the relative roles of peristalsis vs. segmental contraction on the degree of mixing and time scales of nutrient transport to the epithelium using a two-dimensional model of flow and mixing in the small intestine. The model uses the lattice-Boltzmann framework with second-order moving boundary conditions and passive scalar (Sc = 10). Segmental and peristaltic contractions were parameterized using magnetic resonance imaging data from rat models. The Reynolds numbers (1.9), segment lengths (33 mm), max radii (2.75 mm) and occlusion ratios (0.33) were matched for direct comparison. Mixing is quantified by the rate of dispersion of scalar from an initial concentration in the center of the segment. We find that radial mixing is more rapid with segmental than peristaltic motion, that radial dispersion is much more rapid than axial, and that axial is comparable between the motions.
Hu, Kainan; Geng, Shaojuan
2016-01-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e. the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion...
Chen, Li; Kang, Qinjun; Holby, Edward F; Tao, Wen-Quan
2014-01-01
High-resolution porous structures of catalyst layer (CL) with multicomponent in proton exchange membrane fuel cells are reconstructed using a reconstruction method called quartet structure generation set. Characterization analyses of nanoscale structures are implemented including pore size distribution, specific area and phase connectivity. Pore-scale simulation methods based on the lattice Boltzmann method are developed and used to predict the macroscopic transport properties including effective diffusivity and proton conductivity. Nonuniform distributions of ionomer in CL generates more tortuous pathway for reactant transport and greatly reduces the effective diffusivity. Tortuosity of CL is much higher than conventional Bruggeman equation adopted. Knudsen diffusion plays a significant role in oxygen diffusion and significantly reduces the effective diffusivity. Reactive transport inside the CL is also investigated. Although the reactive surface area of non-precious metal catalyst (NPMC) CL is much higher t...
Transport Properties of the Universal Quantum Equation
Institute of Scientific and Technical Information of China (English)
A.I.Arbab
2012-01-01
The universal quantum equation (UQE) is found to describe the transport properties of the quantum particles.This equation describes a wave equation interacting with constant scalar and vector potentials propagating in spacetime.A new transformation that sends the Schr(o)dinger equation with a potential energy V =-1/2mc2 to Dirac's equation is proposed.The Cattaneo telegraph equation as well as a one-dimensional UQE are compatible with our recently proposed generalized continuity equations.Furthermore,a new wave equation resulted from the invariance of the UQE under the post-Galilean transformations is derived.This equation is found to govern a Klein Gordon's particle interacting with a photon-like vector field (ether) whose magnitude is proportional to the particle's mass.
Accurate deterministic solutions for the classic Boltzmann shock profile
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Ying, Jinyong
2016-01-01
The size-modified Poisson-Boltzmann equation (SMPBE) is one important variant of the popular dielectric model, the Poisson-Boltzmann equation (PBE), to reflect ionic size effects in the prediction of electrostatics for a biomolecule in an ionic solvent. In this paper, a new SMPBE hybrid solver is developed using a solution decomposition, the Schwartz's overlapped domain decomposition, finite element, and finite difference. It is then programmed as a software package in C, Fortran, and Python based on the state-of-the-art finite element library DOLFIN from the FEniCS project. This software package is well validated on a Born ball model with analytical solution and a dipole model with a known physical properties. Numerical results on six proteins with different net charges demonstrate its high performance. Finally, this new SMPBE hybrid solver is shown to be numerically stable and convergent in the calculation of electrostatic solvation free energy for 216 biomolecules and binding free energy for a DNA-drug com...
Lahiri, T.; Pal Majumder, T.; Ghosh, N. K.
2014-07-01
Commercialization of ferroelectric liquid crystal displays (FLCDs) suffers from mechanical and electro-convective instabilities. Impurity ions play a pivotal role in the latter case, and therefore we developed a mean-field type model to understand the complex role of space charges, particularly ions in a ferroelectric liquid crystal. Considering an effective ion-chirality relation, we obtained a modified Poisson-Boltzmann equation for ions dissolved into a chiral solvent like the ferroelectric smectic phase. A nonuniform director profile induced by the mean electrostatic potential of the ions is then calculated by solving an Euler-Lagrange equation for a helically twisted smectic state. A combination of effects resulting from molecular chirality and an electrostatically driven twist created by the ions seems to produce this nonuniform fluctuation in the director orientation. Finally, both theoretical and experimental points of view are presented on the prediction of this mean-field model.
Allaire, Gregoire; Dufreche, Jean-Francois; Mikelic, Andro; Piatnitski, Andrey
2013-01-01
This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for Poisson-Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality...
Xiong, Yuan
2014-04-28
Spurious current emerging in the vicinity of phase interfaces is a well-known disadvantage of the lattice Boltzmann equation (LBE) for two-phase flows. Previous analysis shows that this unphysical phenomenon comes from the force imbalance at discrete level inherited in LBE (Guo et al 2011 Phys. Rev. E 83 036707). Based on the analysis of the LBE free of checkerboard effects, in this work we further show that the force imbalance is caused by the different discretization stencils: the implicit one from the streaming process and the explicit one from the discretization of the force term. Particularly, the total contribution includes two parts, one from the difference between the intrinsically discretized density (or ideal gas pressure) gradient and the explicit ones in the force term, and the other from the explicit discretized chemical potential gradients in the intrinsically discretized force term. The former contribution is a special feature of LBE which was not realized previously.
Stable Equilibrium Based on Lévy Statistics:A Linear Boltzmann Equation Approach
Barkai, Eli
2004-06-01
To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass mlaw equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The Boltzmann simplified velocity distribution function equation describing the gas transfer phenomena from various flow regimes will be explored and solved numerically in this study. The discrete velocity ordinate method of the gas kinetic theory is studied and applied to simulate the complex multi-scale flows. Based on the uncoupling technique on molecular movement and colliding in the DSMC method, the gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by extending and applying the unsteady time-splitting method from computational fluid dynamics. The Gauss-type discrete velocity numerical quadrature technique for different Mach number flows is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established to study the three-dimensional complex flows from rarefied transition to continuum regimes. The parallel strategy adapted to the gas-kinetic numerical algorithm is investigated by analyzing the inner parallel degree of the algorithm, and then the HPF parallel processing program is developed. To test the reliability of the present gas-kinetic numerical method, the three-dimensional complex flows around sphere and spacecraft shape with various Knudsen numbers are simulated by HPF parallel computing. The computational results are found in high resolution of the flow fields and good agreement with the theoretical and experimental data. The computing practice has confirmed that the present gas-kinetic algorithm probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view of solving the Boltzmann model equation.
Ender, I A; Flegontova, E Yu; Gerasimenko, A B
2016-01-01
An algorithm for sequential calculation of non-isotropic matrix elements of the collision integral which are necessary for the solution of the non-linear Boltzmann equation by moment method is proposed. Isotropic matrix elements that we believe are known, are starting ones. The procedure is valid for any interaction law and any mass ratio of the colliding particles.
Aspheric surface testing by irradiance transport equation
Shomali, Ramin; Darudi, Ahmad; Nasiri, Sadollah; Asgharsharghi Bonab, Armir
2010-10-01
In this paper a method for aspheric surface testing is presented. The method is based on solving the Irradiance Transport Equation (ITE).The accuracy of ITE normally depends on the amount of the pick to valley of the phase distribution. This subject is investigated by a simulation procedure.
Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten
2016-08-01
The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space-oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multigrid solvers, this approach allows us to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol.
Generalized Poisson-Boltzmann Equation Taking into Account Ionic Interaction and Steric Effects
Institute of Scientific and Technical Information of China (English)
刘新敏; 李航; 李睿; 田锐; 许晨阳
2012-01-01
Generalized Poisson l3oltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density 〈 0.0032 C/dm2, while the ionic interaction is an important effect for electrolyte concentration 〉 0.15 tool/1 in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.
Noise Prevents Singularities in Linear Transport Equations
Fedrizzi, Ennio; Flandoli, Franco
2012-01-01
A stochastic linear transport equation with multiplicative noise is considered and the question of no-blow-up is investigated. The drift is assumed only integrable to a certain power. Opposite to the deterministic case where smooth initial conditions may develop discontinuities, we prove that a certain Sobolev degree of regularity is maintained, which implies H\\"older continuity of solutions. The proof is based on a careful analysis of the associated stochastic flow of characteristics.
Maassen, Jesse; Lundstrom, Mark
2016-03-01
Understanding ballistic phonon transport effects in transient thermoreflectance experiments and explaining the observed deviations from classical theory remains a challenge. Diffusion equations are simple and computationally efficient but are widely believed to break down when the characteristic length scale is similar or less than the phonon mean-free-path. Building on our prior work, we demonstrate how well-known diffusion equations, namely, the hyperbolic heat equation and the Cattaneo equation, can be used to model ballistic phonon effects in frequency-dependent periodic steady-state thermal transport. Our analytical solutions are found to compare excellently to rigorous numerical results of the phonon Boltzmann transport equation. The correct physical boundary conditions can be different from those traditionally used and are paramount for accurately capturing ballistic effects. To illustrate the technique, we consider a simple model problem using two different, commonly used heating conditions. We demonstrate how this framework can easily handle detailed material properties, by considering the case of bulk silicon using a full phonon dispersion and mean-free-path distribution. This physically transparent approach provides clear insights into the nonequilibrium physics of quasi-ballistic phonon transport and its impact on thermal transport properties.
Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan
2016-10-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e., the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion, so it is easy to construct new schemes of higher-order accuracy. To validate the proposed scheme, a one-dimensional shock tube simulation is performed. The numerical results agree with the analytical solutions very well.
Illg, Christian; Haag, Michael; Teeny, Nicolas; Wirth, Jens; Fähnle, Manfred
2016-03-01
Scatterings of electrons at quasiparticles or photons are very important for many topics in solid-state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings, Fermi's golden rule is used to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi's golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.
Lattice Boltzmann Numerical Simulation of a Circular Cylinder
Institute of Scientific and Technical Information of China (English)
冯士德; 赵颖; 郜宪林; 季仲贞
2002-01-01
The lattice Boltzmann equation (LBE) model based on the Boltzmann equation is suitable for the numerical simulation of various flow fields. The fluid dynamics equation can be recovered from the LBE model. However,compared to the Navier-Stokes transport equation, the fluid dynamics equation derived from the LBE model is somewhat different in the viscosity transport term, which contains not only the Navier-Stokes transport equation but also nonsteady pressure and momentum flux terms. The two nonsteady terms can produce the same function as the random stirring force term introduced in the direct numerical or large-eddy vortex simulation of turbulence.Through computation of a circular cylinder, it is verified that the influence of the two nonsteady terms on flow field stability cannot be ignored, which is helpful for the study of turbulence.
Paradis, Hedvig; Andersson, Martin; Sundén, Bengt
2016-08-01
A 3D model at microscale by the lattice Boltzmann method (LBM) is proposed for part of an anode of a solid oxide fuel cell (SOFC) to analyze the interaction between the transport and reaction processes and structural parameters. The equations of charge, momentum, heat and mass transport are simulated in the model. The modeling geometry is created with randomly placed spheres to resemble the part of the anode structure close to the electrolyte. The electrochemical reaction processes are captured at specific sites where spheres representing Ni and YSZ materials are present with void space. This work focuses on analyzing the effect of structural parameters such as porosity, and percentage of active reaction sites on the ionic current density and concentration of H2 using LBM. It is shown that LBM can be used to simulate an SOFC anode at microscale and evaluate the effect of structural parameters on the transport processes to improve the performance of the SOFC anode. It was found that increasing the porosity from 30 to 50 % decreased the ionic current density due to a reduction in the number of reaction sites. Also the consumption of H2 decreased with increasing porosity. When the percentage of active reaction sites was increased while the porosity was kept constant, the ionic current density increased. However, the H2 concentration was slightly reduced when the percentage of active reaction sites was increased. The gas flow tortuosity decreased with increasing porosity.
Energy Technology Data Exchange (ETDEWEB)
Bal, G. [Electricite de France (EDF), Direction des Etudes et Recherches, 92 - Clamart (France)
1997-12-31
Neutron transport in nuclear reactors is well modeled by the linear Boltzmann transport equation. Its resolution is relatively easy but very expensive. To achieve whole core calculations, one has to consider simpler models, such as diffusion or homogeneous transport equations. However, the solutions may become inaccurate in particular situations (as accidents for instance). That is the reason why we wish to solve the equations on small area accurately and more coarsely on the remaining part of the core. It is than necessary to introduce some links between different discretizations or modelizations. In this note, we give some results on the coupling of different discretizations of all degrees of freedom of the integral-differential neutron transport equation (two degrees for the angular variable, on for the energy component, and two or three degrees for spatial position respectively in 2D (cylindrical symmetry) and 3D). Two chapters are devoted to the coupling of discrete ordinates methods (for angular discretization). The first one is theoretical and shows the well posing of the coupled problem, whereas the second one deals with numerical applications of practical interest (the results have been obtained from the neutron transport code developed at the R and D, which has been modified for introducing the coupling). Next, we present the nodal scheme RTN0, used for the spatial discretization. We show well posing results for the non-coupled and the coupled problems. At the end, we deal with the coupling of energy discretizations for the multigroup equations obtained by homogenization. Some theoretical results of the discretization of the velocity variable (well-posing of problems), which do not deal directly with the purposes of coupling, are presented in the annexes. (author). 34 refs.
Lattice Boltzmann Large Eddy Simulation Model of MHD
Flint, Christopher
2016-01-01
The work of Ansumali \\textit{et al.}\\cite{Ansumali} is extended to Two Dimensional Magnetohydrodynamic (MHD) turbulence in which energy is cascaded to small spatial scales and thus requires subgrid modeling. Applying large eddy simulation (LES) modeling of the macroscopic fluid equations results in the need to apply ad-hoc closure schemes. LES is applied to a suitable mesoscopic lattice Boltzmann representation from which one can recover the MHD equations in the long wavelength, long time scale Chapman-Enskog limit (i.e., the Knudsen limit). Thus on first performing filter width expansions on the lattice Boltzmann equations followed by the standard small Knudsen expansion on the filtered lattice Boltzmann system results in a closed set of MHD turbulence equations provided we enforce the physical constraint that the subgrid effects first enter the dynamics at the transport time scales. In particular, a multi-time relaxation collision operator is considered for the density distribution function and a single rel...
Cao, Shanshan; Qin, Guang-You; Wang, Xin-Nian
2016-01-01
A Linearized Boltzmann Transport (LBT) model coupled with hydrodynamical background is established to describe the evolution of jet shower partons and medium excitations in high energy heavy-ion collisions. We extend the LBT model to include both elastic and inelastic processes for light and heavy partons in the quark-gluon plasma. A hybrid model of fragmentation and coalescence is developed for the hadronization of heavy quarks. Within this framework, we investigate how heavy flavor observables depend on various ingredients, such as different energy loss and hadronization mechanisms, the momentum and temperature dependences of the transport coefficients, and the radial flow of the expanding fireball. Our model calculations show good descriptions of $D$ meson suppression and elliptic flow observed at the LHC and RHIC. The prediction for the Pb-Pb collisions at $\\sqrt{s_\\mathrm{NN}}$=5.02 TeV is provided.
Maximal stochastic transport in the Lorenz equations
Energy Technology Data Exchange (ETDEWEB)
Agarwal, Sahil, E-mail: sahil.agarwal@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Wettlaufer, J.S., E-mail: john.wettlaufer@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Departments of Geology & Geophysics, Mathematics and Physics, Yale University, New Haven (United States); Mathematical Institute, University of Oxford, Oxford (United Kingdom); Nordita, Royal Institute of Technology and Stockholm University, Stockholm (Sweden)
2016-01-08
We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh–Bénard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of Souza and Doering [1], but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.
Anomalous Fractional Diffusion Equation for Transport Phenomena
Institute of Scientific and Technical Information of China (English)
QiuhuaZENG; HouqiangLI; 等
1999-01-01
We derive the standard diffusion equation from the continuity equation and by discussing the defectiveness of earlier proposed equations,we get the generalized fractional diffusion equation for anomalous diffusion.
Asymptotic Analysis of Transport Equation in Annulus
Wu, Lei; Yang, Xiongfeng; Guo, Yan
2016-09-01
We consider the diffusive limit of a steady neutron transport equation with one-speed velocity in a two-dimensional annulus. A classical theorem in Bensoussan et al. (Publ Res Inst Math Sci 15:53-157, 1979) states that the solution can be approximated in L^{∞} by the sum of the interior solution and Knudsen layer derived from Milne problem. However, this result was disproved in Wu and Guo (Commun Math Phys 336:1473-1553, 2015) in a plate via a different boundary layer expansion with geometric correction. In this paper, we established the diffusive limit and provide a counterexample to Bensoussan et al. (1979) in non-convex domains.
Blaizot, Jean-Paul; McLerran, Larry
2013-01-01
In this paper, we study the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. In the present study we ignore the effect of the longitudinal expansion, i.e., we restrict ourselves to spatially uniform systems, with spherically symmetric momentum distributions. Furthermore we take into account only elastic scattering, i.e., we neglect inelastic, number changing, processes. We solve the transport equation for various initial conditions that correspond to small or large initial gluon phase-space densities. For a small initial phase-space density, the system evolves towards thermal equilibrium, as expected. For a large enough initial phase-space density the equilibrium state contains a Bose condensate. We present numerical evidence that such over-populated systems rea...
Energy Technology Data Exchange (ETDEWEB)
Mikell, Justin K. [Department of Radiation Physics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); University of Texas Graduate School of Biomedical Sciences at Houston, Houston, Texas (United States); Klopp, Ann H. [Department of Radiation Oncology, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Gonzalez, Graciela M.N. [Department of Biostatistics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Kisling, Kelly D. [Department of Radiation Physics-Patient Care, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); University of Texas Graduate School of Biomedical Sciences at Houston, Houston, Texas (United States); Price, Michael J. [Department of Physics and Astronomy, Louisiana State University and Agricultural and Mechanical College, Baton Rouge, Louisiana, and Mary Bird Perkins Cancer Center, Baton Rouge, Louisiana (United States); Berner, Paula A. [Department of Radiation Physics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Eifel, Patricia J. [Department of Radiation Oncology, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Mourtada, Firas, E-mail: fmourtad@christianacare.org [Department of Radiation Physics-Patient Care, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Department of Experimental Diagnostic Imaging, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Department of Radiation Oncology, Helen F. Graham Cancer Center, Newark, Delaware (United States)
2012-07-01
Purpose: To investigate the dosimetric impact of the heterogeneity dose calculation Acuros (Transpire Inc., Gig Harbor, WA), a grid-based Boltzmann equation solver (GBBS), for brachytherapy in a cohort of cervical cancer patients. Methods and Materials: The impact of heterogeneities was retrospectively assessed in treatment plans for 26 patients who had previously received {sup 192}Ir intracavitary brachytherapy for cervical cancer with computed tomography (CT)/magnetic resonance-compatible tandems and unshielded colpostats. The GBBS models sources, patient boundaries, applicators, and tissue heterogeneities. Multiple GBBS calculations were performed with and without solid model applicator, with and without overriding the patient contour to 1 g/cm{sup 3} muscle, and with and without overriding contrast materials to muscle or 2.25 g/cm{sup 3} bone. Impact of source and boundary modeling, applicator, tissue heterogeneities, and sensitivity of CT-to-material mapping of contrast were derived from the multiple calculations. American Association of Physicists in Medicine Task Group 43 (TG-43) guidelines and the GBBS were compared for the following clinical dosimetric parameters: Manchester points A and B, International Commission on Radiation Units and Measurements (ICRU) report 38 rectal and bladder points, three and nine o'clock, and {sub D2cm3} to the bladder, rectum, and sigmoid. Results: Points A and B, D{sub 2} cm{sup 3} bladder, ICRU bladder, and three and nine o'clock were within 5% of TG-43 for all GBBS calculations. The source and boundary and applicator account for most of the differences between the GBBS and TG-43 guidelines. The D{sub 2cm3} rectum (n = 3), D{sub 2cm3} sigmoid (n = 1), and ICRU rectum (n = 6) had differences of >5% from TG-43 for the worst case incorrect mapping of contrast to bone. Clinical dosimetric parameters were within 5% of TG-43 when rectal and balloon contrast were mapped to bone and radiopaque packing was not overridden
A Multiscale Lattice-Boltzmann Model of Macro-to-Micro Scale Transport Relevant to Gut Function
Wang, Yanxing; Brasseur, James; Banco, Gino
2007-11-01
Nutrient and pharmaceutical absorption in the small intestine involve coupled multiscale transport and mixing processes that span several orders of magnitude. We hypothesize that muscle-induced villi motions generate and control a ``micro-mixing layer'' that couples with macro-scale mixing to enhance molecular transport to and from the epithelium. In this work we developed a 2-D numerical method based on a multigrid strategy within the lattice-Boltzmann framework. We model a macro-scale cavity flow with microscale finger-like villi in pendular motion on the lower surface and evaluate the coupling between macro and micro-scale fluid motions, scalar mixing, and uptake of passive scalar at the villi surface. Preliminary results show that the moving villi can be effective mixers at the micro scale, especially when groups of villi move in a coordinated, but out-of-phase fashion. A time-evolving series of flow recirculation eddies are generated within a micro mixing layer that increase transport of passive scalar from the macro eddy to the surface by advection. Flow parameters such as frequency of pendular motion, spacing between villi and villi grouping, have strong influences on the behaviors of the micro-mixing layer and the efficiency of scalar transport. An extensive analysis is in process to quantify correlation between scalar mixing and flux, details of villi motion, and induced flow patterns.
Raines, Alla
2015-01-01
Numerical solution of non-steady problems of supersonic inflow of a binary mixture of a rarefied gas on a normally posed wall with mirror and diffuse reflection laws is obtained on the basis of the kinetic Boltzmann equation for the model of hard sphere molecules. For calculation of collision integrals we apply the projection method, developed by Tcheremissine for a one-component gas and generalized by the author for a binary gas mixture in the case of cylindrical symmetry. We demonstrate a good qualitative agreement of our results with other authors for one-component gases.
Institute of Scientific and Technical Information of China (English)
段雅丽; 陈先进; 孔令华
2015-01-01
We develop a lattice Boltzmann model for compound Burgers-Korteweg-de Vries ( cBKdV) equation. By properly treating dispersive term uxxx and applying Chapman-Enskog expansion, the governing equation is recovered correctly from lattice Boltzmann equation and local equilibrium distribution functions are obtained. Numerical experiments show that our results agree well with exact solutions and have better numerical accuracy compared with previous numerical results. This hence indicates that the model is satisfactory and efficient.%针对Burgers-Korteweg-de Vries ( cBKdV)复合方程提出一种格子Boltzmann模型。通过恰当地处理色散项uxxx 并运用Chapman-Enskog展开从格子Boltzmann方程推导出宏观方程，从而得到联系微观量与宏观量的局部平衡分布函数。对不同微分方程进行数值实验，数值解与解析解非常吻合，相比于其它数值结果，该格子Boltzmann模型的数值结果更精确，说明该数值模型的高效性。
André, Raíla
2014-01-01
In this work we analyze the dynamics of collisionless self-gravitating systems described by the f(R)-gravity and Boltzmann equation in the weak field approximation, focusing on the Jeans instability for theses systems. The field equations in this approximation were obtained within the Palatini formalism. Through the solution of coupled equations we achieved the collapse criterion for infinite homogeneous fluid and stellar systems, which is given by a dispersion relation. This result is compared with the results of the standard case and the case for f(R)-gravity in metric formalism, in order to see the difference among them. The limit of instability varies according to which theory of gravity is adopted.
Noronha, Jorge
2015-01-01
In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime $\\mathrm{AdS}_{2}\\otimes \\mathrm{S}_{2}$. We further derive explicit analytic expressions for the momentum dependence of the single particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density does not match the equilibrium form. The non-equilibrium contribution to the entropy density is shown to be due to higher order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Thus, in this system the slowly moving hydrodynamic d...
Energy Technology Data Exchange (ETDEWEB)
Urbatsch, Todd James [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2015-06-15
We present an overview of radiation transport, covering terminology, blackbody raditation, opacities, Boltzmann transport theory, approximations to the transport equation. Next we introduce several transport methods. We present a section on Caseology, observing transport boundary layers. We briefly broach topics of software development, including verification and validation, and we close with a section on high energy-density experiments that highlight and support radiation transport.
KdV方程带修正函数的格子Boltzmann模拟%Lattice Boltzmann model with amending function for KdV equation
Institute of Scientific and Technical Information of China (English)
何郁波; 董晓亮; 林晓艳
2012-01-01
A new lattice Boltzmann model with amending-function for KdV equation is presented. Monotonicity and stability of the scheme are analyzed. The conditions of monotonicity are obtained, under which the stability of the scheme is proved in the L1 norm. Applied the proposed scheme, the solutions of KdV equation are simulated, and numerical results agree with the analytical solutions quite well.%采用一种带修正函数的新格子Boltzmann模型模拟了KdV方程,分析了由此得出的迭代格式的单调性和稳定性,得到了格式的单调性条件.在单调性条件下,迭代格式是L1稳定的.数值模拟结果表明该格式是可行的.
A stochastic method of solution of the Parker transport equation
Wawrzynczak, A; Gil, A
2015-01-01
We present the stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Based on the solution of the Parker transport equation we developed models of the short-time variation of the GCR intensity, i.e. the Forbush decrease (Fd) and the 27-day variation of the GCR intensity. Parker transport equation being the Fokker-Planck type equation delineates non-stationary transport of charged particles in the turbulent medium. The presented approach of the numerical solution is grounded on solving of the set of equivalent stochastic differential equations (SDEs). We demonstrate the method of deriving from Parker transport equation the corresponding SDEs in the heliocentric spherical coordinate system for the backward approach. Features indicative the preeminence of the backward approach over the forward is stressed. We compare the outcomes of the stochastic model of the Fd and 27-day variation of the GCR intensity with our former models established by the finite difference method. Both ...
Mendes, Albert C R; Abreu, Everton M C; Neto, Jorge Ananias
2016-01-01
In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled with the electromagnetic fluid and the Dirac's constraints analysis was discussed. A set of first-class constraints fixed by noncovariant gauge condition was obtained. The path integral formalism was used to obtain the partition function for the corresponding higher-derivative Hamiltonian and the Faddeev-Popov ansatz was used to construct an effective Lagrangian. Through the partition function, a Stefan-Boltzmann type law was obtained.
S, Savithiri; Dhar,Purbarun; Pattamatta, Arvind; Das, Sarit K
2015-01-01
Severe contradictions exist between experimental observations and computational predictions regarding natural convective thermal transport in nanosuspensions. The approach treating nanosuspensions as homogeneous fluids in computations has been pin pointed as the major contributor to such contradictions. To fill the void, inter particle and particle fluid interactivities (slip mechanisms), in addition to effective thermophysical properties, have been incorporated within the present formulation...
Discontinuous Galerkin for the Radiative Transport Equation
Guermond, Jean-Luc
2013-10-11
This note presents some recent results regarding the approximation of the linear radiative transfer equation using discontinuous Galerkin methods. The locking effect occurring in the diffusion limit with the upwind numerical flux is investigated and a correction technique is proposed.
Energy Technology Data Exchange (ETDEWEB)
Horsten, N., E-mail: niels.horsten@kuleuven.be; Baelmans, M. [KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300A, 3001 Leuven (Belgium); Dekeyser, W. [ITER Organization, route de Vinon-sur-Verdon, 13067 St. Paul lez Durance Cedex (France); Samaey, G. [KU Leuven, Department of Computer Science, Celestijnenlaan 200A, 3001 Leuven (Belgium)
2016-01-15
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.
A Boltzmann-weighted hopping model of charge transport in organic semicrystalline films
Kwiatkowski, Joe J.
2011-01-01
We present a model of charge transport in polycrystalline electronic films, which considers details of the microscopic scale while simultaneously allowing realistically sized films to be simulated. We discuss the approximations and assumptions made by the model, and rationalize its application to thin films of directionally crystallized poly(3-hexylthiophene). In conjunction with experimental data, we use the model to characterize the effects of defects in these films. Our findings support the hypothesis that it is the directional crystallization of these films, rather than their defects, which causes anisotropic mobilities. © 2011 American Institute of Physics.
Puglisi, Andrea
2015-01-01
This brief offers a concise presentation of granular fluids from the point of view of non-equilibrium statistical physics. The emphasis is on fluctuations, which can be large in granular fluids due to the small system size (the number of grains is many orders of magnitude smaller than in molecular fluids). Firstly, readers will be introduced to the most intriguing experiments on fluidized granular fluids. Then granular fluid theory, which goes through increasing levels of coarse-graining and emerging collective phenomena, is described. Problems and questions are initially posed at the level of kinetic theory, which describes particle densities in full or reduced phase-space. Some answers become clear through hydrodynamics, which describes the evolution of slowly evolving fields. Granular fluctuating hydrodynamics, which builds a bridge to the most recent results in non-equilibrium statistical mechanics, is also introduced. Further and more interesting answers come when the dynamics of a massive intruder are...
Compressible turbulence transport equations for generalized second order closure
Energy Technology Data Exchange (ETDEWEB)
Cloutman, L D
1999-05-01
Progress on the theory of second order closure in turbulence models of various types requires knowledge of the transport equations for various turbulence correlations. This report documents a procedure that provides such equations for a wide variety of turbulence averages for compressible flows of a multicomponent fluid. Generalizing some work by Germano for incompressible flows, we introduce an appropriate extension of his generalized second order correlations and use a generalized mass-weighted averaging procedure to derive transport equations for the correlations. The averaging procedure includes all of the commonly used averages as special cases. The resulting equations provide an internally consistent starting point for future work in developing single-point statistical turbulence transport models for fluid flows. The form invariance of the in-compressible equations also holds for the compressible case, and we discuss some of the closure issues and frequently ignored complications of statistical turbulence models of compressible flows.
Development of interfacial area transport equation - modeling and experimental benchmark
Energy Technology Data Exchange (ETDEWEB)
Ishii, M. [Purdue Univ., West Lafayette, Indiana (United States)
2011-07-01
A dynamic treatment of interfacial area concentration has been studied over the last decade by employing the interfacial area transport equation. When coupled with the two-fluid model, the interfacial area transport equation replaces the flow regime dependent correlations for interfacial area concentration and eliminates potential artificial bifurcation or numerical oscillations stemming from these static correlations. An extensive database has been established to evaluate the model under various two-phase flow conditions. These include adiabatic and heated conditions, vertical and horizontal flow orientations, round, rectangular, annulus and 8×8 rod bundle channel geometries, and normal-gravity and simulated reduced-gravity conditions. This paper reviews the current state-of-the-art in the development of the interfacial area transport equation, available experimental databases and 1D and 3D benchmarking work of the interfacial area transport equation. (author)
UPWIND DISCONTINUOUS GALERKIN METHODS FOR TWO DIMENSIONAL NEUTRON TRANSPORT EQUATIONS
Institute of Scientific and Technical Information of China (English)
袁光伟; 沈智军; 闫伟
2003-01-01
In this paper the upwind discontinuous Galerkin methods with triangle meshes for two dimensional neutron transport equations will be studied.The stability for both of the semi-discrete and full-discrete method will be proved.
SPECTRAL FINITE ELEMENT METHOD FOR A UNSTEADY TRANSPORT EQUATION
Institute of Scientific and Technical Information of China (English)
MeiLiquan
1999-01-01
In this paper,a new numerical method,the coupling method of spherical harmonic function spectral and finite elements,for a unsteady transport equation is dlscussed,and the error analysis of this scheme is proved.
STABILITY OF P2 METHODS FOR NEUTRON TRANSPORT EQUATIONS
Institute of Scientific and Technical Information of China (English)
袁光伟; 沈智军; 沈隆钧; 周毓麟
2002-01-01
In this paper the P2 approximation to the one-group planar neutron transport theory is discussed. The stability of the solutions for P2 equations with general boundary conditions, including the Marshak boundary condition, is proved. Moreover,the stability of the up-wind difference scheme for the P2 equation is demonstrated.
Saturation effects in QCD from linear transport equation
Kutak, Krzysztof
2010-01-01
We show that the GBW saturation model provides an exact solution to the one-dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term.
A Generalized Boltzmann Fokker-Planck Method for Coupled Charged Particle Transport
Energy Technology Data Exchange (ETDEWEB)
Prinja, Anil K
2012-01-09
The goal of this project was to develop and investigate the performance of reduced-physics formulations of high energy charged particle (electrons, protons and heavier ions) transport that are computationally more efficient than not only analog Monte Carlo methods but also the established condensed history Monte Carlo technique. Charged particles interact with matter by Coulomb collisions with target nuclei and electrons, by bremsstrahlung radiation loss and by nuclear reactions such as spallation and fission. Of these, inelastic electronic collisions and elastic nuclear collisions are the dominant cause of energy-loss straggling and angular deflection or range straggling of a primary particle. These collisions are characterized by extremely short mean free paths (sub-microns) and highly peaked, near-singular differential cross sections about forward directions and zero energy loss, with the situation for protons and heavier ions more extreme than for electrons. For this reason, analog or truephysics single-event Monte Carlo simulation, while possible in principle, is computationally prohibitive for routine calculation of charged particle interaction phenomena.
Lattice Boltzmann model for nanofluids
Energy Technology Data Exchange (ETDEWEB)
Xuan Yimin; Yao Zhengping [Nanjing University of Science and Technology, School of Power Engineering, Nanjing (China)
2005-01-01
A nanofluid is a particle suspension that consists of base liquids and nanoparticles and has great potential for heat transfer enhancement. By accounting for the external and internal forces acting on the suspended nanoparticles and interactions among the nanoparticles and fluid particles, a lattice Boltzmann model is proposed for simulating flow and energy transport processes inside the nanofluids. First, we briefly introduce the conventional lattice Boltzmann model for multicomponent systems. Then, we discuss the irregular motion of the nanoparticles and inherent dynamic behavior of nanofluids and describe a lattice Boltzmann model for simulating nanofluids. Finally, we conduct some calculations for the distribution of the suspended nanoparticles. (orig.)
Macroscopic heat transport equations and heat waves in nonequilibrium states
Guo, Yangyu; Jou, David; Wang, Moran
2017-03-01
Heat transport may behave as wave propagation when the time scale of processes decreases to be comparable to or smaller than the relaxation time of heat carriers. In this work, a generalized heat transport equation including nonlinear, nonlocal and relaxation terms is proposed, which sums up the Cattaneo-Vernotte, dual-phase-lag and phonon hydrodynamic models as special cases. In the frame of this equation, the heat wave propagations are investigated systematically in nonequilibrium steady states, which were usually studied around equilibrium states. The phase (or front) speed of heat waves is obtained through a perturbation solution to the heat differential equation, and found to be intimately related to the nonlinear and nonlocal terms. Thus, potential heat wave experiments in nonequilibrium states are devised to measure the coefficients in the generalized equation, which may throw light on understanding the physical mechanisms and macroscopic modeling of nanoscale heat transport.
Directory of Open Access Journals (Sweden)
K. Banoo
1998-01-01
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.
Energy Technology Data Exchange (ETDEWEB)
Fidler, Christian
2011-12-16
Polarisation and Nongaussianity are expected to play a central role in future studies of the cosmic microwave background radiation. Polarisation can be split into a divergence-like E-mode and a curl-like B-mode, of which the later can only be induced by primordial gravitational waves (tensor fluctuations of the metric) at leading order. Nongaussianity is not generated at first order and is directly proportional to the primordial Nongaussianity of inflation. Thus B-mode polarisation and Nongaussianity constrain inflation models directly. While E-mode polarisation has already been detected and is being observed with increasing precision, B-mode polarisation and Nongaussianity remains elusive. The absence of B-mode polarisation when the primordial fluctuations are purely scalar holds, however, only in linear perturbation theory. B-mode polarisation is also generated from scalar sources in second order, which may constitute an important background to the search for primordial gravitational waves. While such an effect would naturally be expected to be relevant at tensor-to-scalar ratios of order 10{sup -5}, which is the size of perturbations in the microwave background, only a full second order calculation can tell whether there are no enhancements. For Nongaussianity the situation is analogous: At second order intrinsic Nongaussianities are induced to the spectrum, which may be an important background to the primordial Nongaussianity. After the full second-order Boltzmann equations for the cosmological evolution of the polarised radiation distribution have become available, I focused on the novel sources to B-mode polarisation that appear in the second-order collision term, which have not been calculated before. In my PHD thesis I developed a numerical code, which solves the second order Boltzmann hierarchy and calculates the C{sub l}{sup BB}-spectrum.
Gao, Jinfang; Xing, Huilin; Tian, Zhiwei; Pearce, Julie K.; Sedek, Mohamed; Golding, Suzanne D.; Rudolph, Victor
2017-01-01
Injection of CO2 subsurface may lead to chemical reactivity of rock where CO2 is dissolved in groundwater. This process can modify pore networks to increase or decrease porosity through mineral dissolution and precipitation. A lattice Boltzmann (LB) based computational model study on the pore scale reactive transport in three dimensional heterogeneous porous media (sandstone consisting of both reactive and non-reactive minerals) is described. This study examines how fluid transport in porous materials subject to reactive conditions is affected by unsteady state local reactions and unstable dissolution fronts. The reaction of a calcite cemented core sub-plug from the Hutton Sandstone of the Surat Basin, Australia, is used as a study case. In particular, the work studies the interaction of acidic fluid (an aqueous solution with an elevated concentration of carbonic acid) with reactive (e.g. calcite) and assumed non-reactive (e.g. quartz) mineral surfaces, mineral dissolution and mass transfer, and resultant porosity change. The proposed model is implemented in our custom LBM code and suitable for studies of multiple mineral reactions with disparate reaction rates. A model for carbonic acid reaction with calcite cemented sandstone in the CO2-water-rock system is verified through laboratory experimental data including micro-CT characterization before and after core reaction at reservoir conditions. The experimentally validated model shows: (1) the dissolution of calcite cement forms conductive channels at the pore scale, and enables the generation of pore throats and connectivity; (2) the model is able to simulate the reaction process until the reaction equilibrium status is achieved (around 1440 days); (3) calcite constituting a volume of around 9.6% of the whole core volume is dissolved and porosity is consequently increased from 1.1% to 10.7% on reaching equilibrium; (4) more than a third of the calcite (constituting 7.4% of the total core volume) is unaffected
Mezzacappa, A; Liebendörfer, M; Messer, O E; Hix, W R; Thielemann, F K; Bruenn, S W
2001-03-05
With exact three-flavor Boltzmann neutrino transport, we simulate the stellar core collapse, bounce, and postbounce evolution of a 13M star in spherical symmetry, the Newtonian limit, without invoking convection. In the absence of convection, prior spherically symmetric models, which implemented approximations to Boltzmann transport, failed to produce explosions. We consider exact transport to determine if these failures were due to the transport approximations made and to answer remaining fundamental questions in supernova theory. The model presented here is the first in a sequence of models beginning with different progenitors. In this model, a supernova explosion is not obtained.
Reciprocal Symmetric Boltzmann Function and Unified Boson-Fermion Statistics
2007-01-01
The differential equation for Boltzmann's function is replaced by the corresponding discrete finite difference equation. The difference equation is, then, symmetrized so that the equation remains invariant when step d is replaced by -d. The solutions of this equation come in Boson-Fermion pairs. Reciprocal symmetric Boltzmann's function, thus, unifies both Bosonic and Fermionic distributions.
关于Boltzmann方程的空间均匀的ES模型%On the Spatially Homogeneous ES Model of the Boltzmann Equation
Institute of Scientific and Technical Information of China (English)
张显文; 胡适耕; 张诚坚
2006-01-01
We discuss the solution properties of the spatially homogeneous ellipsoidal statistical model (ES model) of the Boltzmann equation with Prandtl number Pr ∈ [2/3,∞). First, we establish the existence and uniqueness result. Second, the trend towards equilibrium and the Maxwellian lower bound estimates of the solution are established. Finally, we prove the entropy identity and obtain dissipation of the entropy of this equation.%在Prandtl数Pr∈[2/3,∞)的情况下,我们讨论了Boltzmann方程的空间均匀的椭圆统计模型.首先,我们建立了解的存在唯一性.其次,我们证明了该解收敛到平衡态并给出了其Maxwell分布型的下界估计.最后,我们给出了熵等式从而证明了该方程的熵是衰减的.
Boltzmann-Langevin one-body dynamics for fermionic systems
Directory of Open Access Journals (Sweden)
Napolitani P.
2012-07-01
Full Text Available A full implementation of the Boltzmann-Langevin equation for fermionic systems is introduced in a transport model for dissipative collisions among heavy nuclei. Fluctuations are injected in phase space and not, like in more conventional approaches, as a projection on suitable subspaces. The advantage of this model is to be specifically adapted to describe processes characterised by instabilities, like the formation of fragments from a hot nuclear system, and by dissipation, like the transparency in nucleus-nucleus collisions.
Energy Technology Data Exchange (ETDEWEB)
Blaizot, Jean-Paul [Institut de Physique Théorique, CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette (France); Liao, Jinfeng [Physics Dept. and CEEM, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); McLerran, Larry [Physics Dept., Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, China Central Normal University, Wuhan (China)
2014-11-15
To understand the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions, is an important and challenging problem. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. The role of Bose statistical factors in amplifying the rapid growth of the population of the soft modes is essential. With these factors properly taken into account, one finds that elastic scattering alone provides an efficient mechanism for populating soft modes, and in fact leads to rapid infrared local thermalization. Furthermore, recent developments suggest that high initial overpopulation plays a key role and may lead to dynamical Bose–Einstein condensation. The kinetics of condensation is an interesting problem in itself. By solving the transport equation for initial conditions with a large enough initial phase-space density the equilibrium state contains a Bose condensate, and we present numerical evidence that such over-occupied systems reach the onset of Bose–Einstein condensation in a finite time. It is also found that the approach to condensation is characterized by a scaling behavior. Finally we discuss a number of extensions of the present study.
Energy Technology Data Exchange (ETDEWEB)
Blaizot, Jean-Paul [Institut de Physique Théorique, CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette (France); Liao, Jinfeng [Physics Department and Center for Exploration of Energy and Matter, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); McLerran, Larry [RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, China Central Normal University, Wuhan (China)
2013-12-20
In this paper, we study the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. In the present study we ignore the effect of the longitudinal expansion, i.e., we restrict ourselves to spatially uniform systems, with spherically symmetric momentum distributions. Furthermore we take into account only elastic scattering, i.e., we neglect inelastic, number changing, processes. We solve the transport equation for various initial conditions that correspond to small or large initial gluon phase-space densities. For a small initial phase-space density, the system evolves towards thermal equilibrium, as expected. For a large enough initial phase-space density the equilibrium state contains a Bose condensate. We present numerical evidence that such over-populated systems reach the onset of Bose–Einstein condensation in a finite time. The approach to condensation is characterized by a scaling behavior that we briefly analyze.
Properties of an affine transport equation and its holonomy
Vines, Justin; Nichols, David A.
2016-10-01
An affine transport equation was used recently to study properties of angular momentum and gravitational-wave memory effects in general relativity. In this paper, we investigate local properties of this transport equation in greater detail. Associated with this transport equation is a map between the tangent spaces at two points on a curve. This map consists of a homogeneous (linear) part given by the parallel transport map along the curve plus an inhomogeneous part, which is related to the development of a curve in a manifold into an affine tangent space. For closed curves, the affine transport equation defines a "generalized holonomy" that takes the form of an affine map on the tangent space. We explore the local properties of this generalized holonomy by using covariant bitensor methods to compute the generalized holonomy around geodesic polygon loops. We focus on triangles and "parallelogramoids" with sides formed from geodesic segments. For small loops, we recover the well-known result for the leading-order linear holonomy (˜ Riemann × area), and we derive the leading-order inhomogeneous part of the generalized holonomy (˜ Riemann × area^{3/2}). Our bitensor methods let us naturally compute higher-order corrections to these leading results. These corrections reveal the form of the finite-size effects that enter into the holonomy for larger loops; they could also provide quantitative errors on the leading-order results for finite loops.
A Nonlinera Krylov Accelerator for the Boltzmann k-Eigenvalue Problem
Calef, Matthew T; Warsa, James S; Berndt, Markus; Carlson, Neil N
2011-01-01
We compare variants of Anderson Mixing with the Jacobian-Free Newton-Krylov and Broyden methods applied to the k-eigenvalue formulation of the linear Boltzmann transport equation. We present evidence that one variant of Anderson Mixing finds solutions in the fewest number of iterations. We examine and strengthen theoretical results of Anderson Mixing applied to linear problems.
The numerical solution of the vorticity transport equation
Dennis, S C R
1973-01-01
A method of approximating the two-dimensional vorticity transport equation in which the matrix associated with the difference equations is diagonally dominant and the truncation error is the same as that of the fully central-difference approximation, is discussed. An example from boundary layer theory is given by calculating the viscous stagnation point flow at the nose of a cylinder. Some new solutions of the Navier-Stokes equations are obtained for symmetrical flow past a flat plate of finite length. (16 refs).
Energy Technology Data Exchange (ETDEWEB)
Kawakami, H.; Urabe, J.; Yukimura, K. (Doshisha Univ., Kyoto (Japan))
1991-03-20
In a discharge excitation rare gas halide excima laser, uniform generation and stable maintenance of the excited discharge determines the laser characteristics. In this report, an approximate solution was obtained on the Boltzmann equation (frequently used for the theoretical analysis of this laser) to examine the nature of the solution. By optimizing the conversion of the variables, calculation of an electron swarm parameter in the hitherto uncertain range of the low conversion electric field was made possible, giving a generation mechanism of the uncertainty of the excited dischareg. The results are summarized as below. (1) The Boltzmann equation gives a linear solution for a logarithmic value of an electron energy in the range of low conversion electric field. (2) Time-wise responce ability between the measured voltage, current characteristics of the excitation discharge was clarified and the attachment and ionization coefficients calculated by Boltzmann equation. (3) Dependency of the attachment coefficient on the partial pressure of fluorine and kripton was examined, and the attachment coefficient was found to increase with the increase of the partial pressure for the both cases. 20 refs., 9 figs., 2 tabs.
Institute of Scientific and Technical Information of China (English)
Deng Yun-Kun; Xiao Deng-Ming
2013-01-01
The electron swarm parameters including the density-normalized effective ionization coefficients (α-η)/N and the electron drift velocities Ve are calculated for a gas mixture of CF3I with N2 and CO2 by solving the Boltzmann equation in the condition of a steady-state Townsend (SST) experiment.The overall density-reduced electric field strength is from 100 Td to 1000 Td (1 Td =10-17 V·cm2),while the CF3I content k in the gas mixture can be varied over the range from 0％ to 100％.From the variation of (α-η)/N with the CF3I mixture ratio k,the limiting field strength (E/N)lim for each CF3I concentration is derived.It is found that for the mixtures with 70％ CF3I,the values of (E/N)lim are essentially the same as that for pure SF6.Additionally,the global warming potential (GWP) and the liquefaction temperature of the gas mixtures are also taken into account to evaluate the possibility of application in the gas insulation of power equipment.
Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli
2016-10-01
The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter kmaxη >3 , where kmax is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of kmaxη >1 for the pseudospectral method and kmaxη >2 for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.
Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli
2016-10-01
The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter k_{max}η>3, where k_{max} is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of k_{max}η>1 for the pseudospectral method and k_{max}η>2 for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.
Numerical Solution of the Equation of Electron Transport in Matter
Golovin, A I
2002-01-01
One introduces a numerical approach to solve equation of fast electron transport in a matter in plane and spherical geometry with regard to fluctuations of energy losses and generation of secondary electrons. Calculation results are shown to be in line with the experimental data. One compared the introduced approach with the method of moments
Transport modelling in coastal waters using stochastic differential equations
Charles, W.M.
2007-01-01
In this thesis, the particle model that takes into account the short term correlation behaviour of pollutants dispersion has been developed. An efficient particle model for sediment transport has been developed. We have modified the existing particle model by adding extra equations for the suspensio
MSSM electroweak baryogenesis and flavour mixing in transport equations
Energy Technology Data Exchange (ETDEWEB)
Konstandin, Thomas [Institut fuer Theoretische Physik, Heidelberg University, Philosophenweg 16, D-69120 Heidelberg (Germany)]. E-mail: t.konstandin@thphys.uni-heidelberg.de; Prokopec, Tomislav [Institute for Theoretical Physics (ITF) and Spinoza Institute, Utrecht University, Leuvenlaan 4, Postbus 80.195, 3508 TD Utrecht (Netherlands)]. E-mail: t.prokopec@phys.uu.nl; Schmidt, Michael G. [Institut fuer Theoretische Physik, Heidelberg University, Philosophenweg 16, D-69120 Heidelberg (Germany)]. E-mail: m.g.schmidt@thphys.uni-heidelberg.de; Seco, Marcos [Institut fuer Theoretische Physik, Heidelberg University, Philosophenweg 16, D-69120 Heidelberg (Germany)]. E-mail: m.seco@thphys.uni-heidelberg.de
2006-03-20
We make use of the formalism of [T. Konstandin, et al., hep-ph/0410135], and calculate the chargino-mediated baryogenesis in the Minimal Supersymmetric Standard Model. The formalism makes use of a gradient expansion of the Kadanoff-Baym equations for mixing fermions. For illustrative purposes, we first discuss the semiclassical transport equations for mixing bosons in a space-time-dependent Higgs background. To calculate the baryon asymmetry, we solve a standard set of diffusion equations, according to which the chargino asymmetry is transported to the top sector, where it biases sphaleron transitions. At the end we make a qualitative and quantitative comparison of our results with the existing work. We find that the production of the baryon asymmetry of the universe by CP-violating currents in the chargino sector is strongly constrained by measurements of electric dipole moments.
Energy Technology Data Exchange (ETDEWEB)
Wang, Chi-Jen [Iowa State Univ., Ames, IA (United States)
2013-01-01
In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.
Transport equations for a general class of evolution equations with random perturbations
Guo, Maozheng; Wang, Xiao-Ping
1999-10-01
We derive transport equations from a general class of equations of form iut=H(X,D)u+V(X,D)u where H(X,D) and V(X,D) are pseudodifferential operators (Weyl operator) with symbols H(x,k) and V(x,k), where H(x,k) being polynomial in k and smooth in x,V(x,k) is a mean zero random function and is stationary in space variable. We also consider system of equations in the above form. Such equations cover many of the equations that arise in wave propagations, such as those considered in a paper by Ryzhik, Papanicolaou, and Keller [Wave Motion 24, 327-370 (1996)]. Our results generalize those by Ryzhik, Papanicolau, and Keller.
Lattice Boltzmann method with the cell-population equilibrium
Institute of Scientific and Technical Information of China (English)
Zhou Xiao-Yang; Cheng Bing; Shi Bao-Chang
2008-01-01
The central problem of the lattice Boltzmann method (LBM) is to construct a discrete equilibrium.In this paper,a multi-speed 1D cell-model of Boltzmann equation is proposed,in which the cell-population equilibrium,a direct nonnegative approximation to the continuous Maxwellian distribution,plays an important part.By applying the explicit one-order Chapman-Enskog distribution,the model reduces the transportation and collision,two basic evolution steps in LBM,to the transportation of the non-equilibrium distribution.Furthermore,1D dam-break problem is performed and the numerical results agree well with the analytic solutions.
The Einstein-Boltzmann system and positivity
Lee, Ho
2012-01-01
The Einstein-Boltzmann system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and by Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the Einstein-Boltzmann system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the Einstein-Boltzmann system so as to include scattering kernels which are physically well-motivated.
Guo, Pengfei; Guan, Yong; Liu, Gang; Liang, Zhiting; Liu, Jianhong; Zhang, Xiaobo; Xiong, Ying; Tian, Yangchao
2016-09-01
This work reports an investigation of the impact of microstructure on the performance of solid oxide fuel cells (SOFC) composed of nickel yttria-stabilized zirconia (Ni YSZ). X-ray nano computed tomography (nano-CT) was used to obtain three-dimensional (3D) models of Ni-YSZ composite anode samples subjected to different thermal cycles. Key parameters, such as triple phase boundary (TPB) density, were calculated using 3D reconstructions. The electrochemical reaction occurring at active-TPB was modeled by the Lattice Boltzmann Method for simulation of multi-component mass transfer in porous anodes. The effect of different electrode geometries on the mass transfer and the electrochemical reaction in anodes was studied by TPB distributions measured by nano CT for samples subjected to different thermal cycles. The concentration polarization and the activation polarization were estimated respectively. The results demonstrate that a combined approach involving nano-CT experiments in conjunction with simulations of gas transport and electrochemical reactions using the Lattice Boltzmann method can be used to better understand the relationship between electrode microstructure and performance of nickel yttria-stabilized zirconia anodes.
Moment transport equations and their application to the perturbed universe
Sierra, Carlos A
2015-01-01
There are many inflationary models that allow the formation of the large-scale structure of the observable universe. The non-gaussianity parameter $f_{NL}$ is a useful tool to discriminate among these cosmological models when comparing the theoretical predictions with the satellite survey results like those from WMAP and Planck. The goal of this proceeding contribution is to review the moment transport equations methodology and the subsequent calculation of the $f_{NL}$ parameter.
Flavour Covariant Transport Equations: an Application to Resonant Leptogenesis
Dev, P S Bhupal; Pilaftsis, Apostolos; Teresi, Daniele
2014-01-01
We present a fully flavour-covariant formalism for transport phenomena, by deriving Markovian master equations that describe the time-evolution of particle number densities in a statistical ensemble with arbitrary flavour content. As an application of this general formalism, we study flavour effects in a scenario of resonant leptogenesis (RL) and obtain the flavour-covariant evolution equations for heavy-neutrino and lepton number densities. This provides a complete and unified description of RL, capturing three relevant physical phenomena: (i) the resonant mixing between the heavy-neutrino states, (ii) coherent oscillations between different heavy-neutrino flavours, and (iii) quantum decoherence effects in the charged-lepton sector. To illustrate the importance of this formalism, we numerically solve the flavour-covariant rate equations for a minimal RL model and show that the total lepton asymmetry can be enhanced up to one order of magnitude, as compared to that obtained from flavour-diagonal or partially ...
Error transport equation boundary conditions for the Euler and Navier-Stokes equations
Phillips, Tyrone S.; Derlaga, Joseph M.; Roy, Christopher J.; Borggaard, Jeff
2017-02-01
Discretization error is usually the largest and most difficult numerical error source to estimate for computational fluid dynamics, and boundary conditions often contribute a significant source of error. Boundary conditions are described with a governing equation to prescribe particular behavior at the boundary of a computational domain. Boundary condition implementations are considered sufficient when discretized with the same order of accuracy as the primary governing equations; however, careless implementations of boundary conditions can result in significantly larger numerical error. Investigations into different numerical implementations of Dirichlet and Neumann boundary conditions for Burgers' equation show a significant impact on the accuracy of Richardson extrapolation and error transport equation discretization error estimates. The development of boundary conditions for Burgers' equation shows significant improvements in discretization error estimates in general and a significant improvement in truncation error estimation. The latter of which is key to accurate residual-based discretization error estimation. This research investigates scheme consistent and scheme inconsistent implementations of inflow and outflow boundary conditions up to fourth order accurate and a formulation for a slip wall boundary condition for truncation error estimation are developed for the Navier-Stokes and Euler equations. The scheme consistent implementation resulted in much smoother truncation error near the boundaries and more accurate discretization error estimates.
Energy Technology Data Exchange (ETDEWEB)
Uchida, S.; Sugawara, H.; Ventzek, P.; Sakai, Y. [Hokkaido University, Sapporo (Japan)
1998-06-01
Xe/Ne plasmas are important for plasma display panels and VUV light sources. However, reactions between electrons and excited particles in the mixtures are so complicated that influence of the reactions on the plasma properties is not understood well. In this work, taking account of reactions through which electrons are produced, such as cumulative and Penning ionization, and of transition between excited levels, the electron and excited particle properties in Xe/Ne plasmas are calculated using the Boltzmann equation. The ionization coefficient and electron drift velocity agreed with experimental data. The influence of laser absorption in Xe/Ne plasmas on the plasma properties is also discussed. 25 refs., 15 figs.
Viscous QCD matter in a hybrid hydrodynamic+Boltzmann approach
Song, Huichao; Heinz, Ulrich W
2010-01-01
A hybrid transport approach for the bulk evolution of viscous QCD matter produced in ultra-relativistic heavy-ion collisions is presented. The expansion of the dense deconfined phase of the reaction is modeled with viscous hydrodynamics while the dilute late hadron gas stage is described microscopically by the Boltzmann equation. The advantages of such a hybrid approach lie in the improved capability of handling large dissipative corrections in the late dilute phase of the reaction, including a realistic treatment of the non-equilibrium hadronic chemistry and kinetic freeze-out. By varying the switching temperature at which the hydrodynamic output is converted to particles for further propagation with the Boltzmann cascade we test the ability of the macroscopic hydrodynamic approach to emulate the microscopic evolution during the hadronic stage and extract the temperature dependence of the effective shear viscosity of the hadron resonance gas produced in the collision. We find that the extracted values depend...
Morales-Casique, E.; Lezama-Campos, J. L.; Guadagnini, A.; Neuman, S. P.
2013-05-01
Modeling tracer transport in geologic porous media suffers from the corrupt characterization of the spatial distribution of hydrogeologic properties of the system and the incomplete knowledge of processes governing transport at multiple scales. Representations of transport dynamics based on a Fickian model of the kind considered in the advection-dispersion equation (ADE) fail to capture (a) the temporal variation associated with the rate of spreading of a tracer, and (b) the distribution of early and late arrival times which are often observed in field and/or laboratory scenarios and are considered as the signature of anomalous transport. Elsewhere we have presented exact stochastic moment equations to model tracer transport in randomly heterogeneous aquifers. We have also developed a closure scheme which enables one to provide numerical solutions of such moment equations at different orders of approximations. The resulting (ensemble) average and variance of concentration fields were found to display a good agreement against Monte Carlo - based simulation results for mildly heterogeneous (or well-conditioned strongly heterogeneous) media. Here we explore the ability of the moment equations approach to describe the distribution of early arrival times and late time tailing effects which can be observed in Monte-Carlo based breakthrough curves (BTCs) of the (ensemble) mean concentration. We show that BTCs of mean resident concentration calculated at a fixed space location through higher-order approximations of moment equations display long tailing features of the kind which is typically associated with anomalous transport behavior and are not represented by an ADE model with constant dispersive parameter, such as the zero-order approximation.
Energy Technology Data Exchange (ETDEWEB)
Bal, G.
1995-07-01
To achieve whole core calculations of the neutron transport equation, we have to follow this 2 step method: space and energy homogenization of the assemblies; resolution of the homogenized equation on the whole core. However, this is no more valid when accidents occur (for instance depressurization causing locally strong heterogeneous media). One solution consists then in coupling two kinds of resolutions: a fine computation on the damaged cell (fine mesh, high number of energy groups) coupled with a coarse one everywhere else. We only deal here with steady state solutions (which already live in 6D spaces). We present here two such methods: The coupling by transmission of homogenized sections and the coupling by transmission of boundary conditions. To understand what this coupling is, we first restrict ourselves to 1D with respect to space in one energy group. The first two chapters deal with a recall of basic properties of the neutron transport equation. We give at chapter 3 some indications of the behaviour of the flux with respect to the cross sections. We present at chapter 4 some couplings and give some properties. Chapter 5 is devoted to a presentation of some numerical applications. (author). 9 refs., 7 figs.
Classical transport equations for burning gas-metal plasmas
Molvig, Kim; Simakov, Andrei N.; Vold, Erik L.
2014-09-01
Thermonuclear inertial confinement fusion plasmas confined by a heavy metal shell may be subject to the mixing of metal into the gas with a resulting degradation of fusion yield. Classical plasma diffusion driven by a number of gradients can provide a physical mechanism to produce atomic mix, possibly in concert with complex hydrodynamic structures and/or turbulence. This paper gives a derivation of the complete dissipative plasma hydrodynamics equations from kinetic theory, for a binary ionic mixture plasma consisting of electrons, e, a light (hydrogenic gas) ion species, i, and a heavy, high ZI plasma metal species, I. A single mean ionization state for the heavy metal, ZI, is assumed to be provided by some independent thermodynamic model of the heavy metal Z I = Z I ( n i , n I , T e ). The kinetic equations are solved by a generalized Chapman-Enskog expansion that assumes small Knudsen numbers for all species: N K e ≡ λ e / L ≪ 1 , N K i ≡ λ i / L ≪ 1. The small electron to ion mass ratio, m e / m i ≪ 1, is utilized to account for electron-ion temperature separation, T e ≠ T i, and to decouple the electron and ion transport coefficient calculations. This produces a well ordered perturbation theory for the electrons, resulting in the well known "Spitzer" problem of Spitzer and collaborators and solved independently by Braginskii. The formulation in this paper makes clear the inherent symmetry of the transport and gives an analytic solution for all values of the effective charge Z eff, including Z eff replaces the Z eff of the electron problem, but has an extended domain, 0≤ Δ I < ∞, to cover all mixture fractions from the pure gas to the pure metal plasma. The extension of the Spitzer problem to include this extended domain is given in this work. The resulting transport equations for the binary gas-metal plasma mixture are complete and accurate through second order. All transport coefficients are provided in analytic form.
A unified transport equation for both cosmic rays and thermal particles
Williams, L. L.; Schwadron, N.; Jokipii, J. R.; Gombosi, T. I.
1993-01-01
We present a unified transport equation that is valid for particles of all energies if the particle mean free paths are much smaller than macroscopic fluid length scales. If restricted to particles with random speeds much greater than fluid flow speeds, this equation reduces to the previously discussed extended cosmic-ray transport equation. It is significant that this allows one to describe the acceleration of particles from thermal energies to cosmic-ray energies using one transport equation. This is in contrast to previous transport equations (the Parker equation and the extended cosmic-ray transport equation), which were restricted to fast particles. The close connection to the extended cosmic-ray transport equation is demonstrated.
Neglected transport equations: extended Rankine-Hugoniot conditions and J -integrals for fracture
Davey, K.; Darvizeh, R.
2016-09-01
Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine-Hugoniot conditions for fracture are established along with extended forms of J -integrals.
Lattice Boltzmann approach for complex nonequilibrium flows.
Montessori, A; Prestininzi, P; La Rocca, M; Succi, S
2015-10-01
We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion.
Flavour covariant transport equations: An application to resonant leptogenesis
Directory of Open Access Journals (Sweden)
P.S. Bhupal Dev
2014-09-01
Full Text Available We present a fully flavour-covariant formalism for transport phenomena, by deriving Markovian master equations that describe the time-evolution of particle number densities in a statistical ensemble with arbitrary flavour content. As an application of this general formalism, we study flavour effects in a scenario of resonant leptogenesis (RL and obtain the flavour-covariant evolution equations for heavy-neutrino and lepton number densities. This provides a complete and unified description of RL, capturing three distinct physical phenomena: (i the resonant mixing between the heavy-neutrino states, (ii coherent oscillations between different heavy-neutrino flavours, and (iii quantum decoherence effects in the charged-lepton sector. To illustrate the importance of this formalism, we numerically solve the flavour-covariant rate equations for a minimal RL model and show that the total lepton asymmetry can be enhanced by up to one order of magnitude, as compared to that obtained from flavour-diagonal or partially flavour off-diagonal rate equations. Thus, the viable RL model parameter space is enlarged, thereby enhancing further the prospects of probing a common origin of neutrino masses and the baryon asymmetry in the Universe at the LHC, as well as in low-energy experiments searching for lepton flavour and number violation. The key new ingredients in our flavour-covariant formalism are rank-4 rate tensors, which are required for the consistency of our flavour-mixing treatment, as shown by an explicit calculation of the relevant transition amplitudes by generalizing the optical theorem. We also provide a geometric and physical interpretation of the heavy-neutrino degeneracy limits in the minimal RL scenario. Finally, we comment on the consistency of various suggested forms for the heavy-neutrino self-energy regulator in the lepton-number conserving limit.
Energy Technology Data Exchange (ETDEWEB)
Verdu, G. [Departamento de Ingenieria Quimica Y Nuclear, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain); Capilla, M.; Talavera, C. F.; Ginestar, D. [Dept. of Nuclear Engineering, Departamento de Matematica Aplicada, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain)
2012-07-01
PL equations are classical high order approximations to the transport equations which are based on the expansion of the angular dependence of the angular neutron flux and the nuclear cross sections in terms of spherical harmonics. A nodal collocation method is used to discretize the PL equations associated with a neutron source transport problem. The performance of the method is tested solving two 1D problems with analytical solution for the transport equation and a classical 2D problem. (authors)
Simplified P$_N$ Equations for Nonclassical Transport with Isotropic Scattering
Vasques, R
2016-01-01
A nonclassical diffusion approximation has been previously derived for the the one-speed nonclassical transport equation with isotropic scattering. In this paper we use an asymptotic analysis to derive more accurate diffusion approximations to the nonclassical transport equation. If the free-path distribution is given by an exponential (classical transport), these approximations reduce to the simplified P$_N$ (SP$_N$) equations; therefore, they are labeled nonclassical SP$_N$ equations.
Schaap, M. G.
2013-12-01
This DOE-funded study is a collaboration between Oregon State University (led by Dr. Dorthe Wildenschild) and the University of Arizona to investigate pore-scale aspects of capillary trapping to enhance the efficiency of geological CO2 sequestration in deep saline aquifers where super-critical conditions prevail. Compared to most current reservoir-scale studies, our research takes several steps back in scale to observe and model trapping at the pore-scale using a combination of computed micro-tomography imaging (performed by OSU) and multi-phase/multi-component lattice Boltzmann (LB) simulations (carried out by UA). The main objective is to quantify how pore-scale mechanisms translate into continuum scale properties that can subsequently support improved modelling of sequestration at large spatio-temporal scales. For the purposes of this project it is important to correctly simulate the physical conditions under which super-critical CO2 will be present after injection into the host rock. In practice this means that the LB model should be able to handle the pressures (P), densities (ρ), temperatures (T) that prevail in deep geological media. A logical way of dealing with is is to combine a single-component LB model with and Equation of State (EOS) that describes the physical interrelations among P, ρ and T (Yuan and Scheafer, 2006). Previously, we showed that the Peng-Robinson (PR) EOS provides an excellent fit to super-critical conditions for the pure CO2 system. However, simulating pure-CO2 systems is not sufficient as the super-critical CO2 will co-exist (and interact) with brine present in the saline aquifers. In effect this means that we need to simulate multi-component systems: one phase being the super-critical CO2, the other phase being a brine of varying salinity. Previously, we have used used a Shan-Chen-type model (Shan Chen, 1993, 1994) as modified by Martys and Chen (1996) for simplified capillari pressure dominated air-water systems in porous media
Transport coefficients from the boson Uehling-Uhlenbeck equation.
Gust, Erich D; Reichl, L E
2013-04-01
Expressions for the bulk viscosity, shear viscosity, and thermal conductivity of a quantum degenerate Bose gas above the critical temperature for Bose-Einstein condensation are derived using the Uehling-Uhlenbeck kinetic equation. For contact potentials and hard sphere interactions, the eigenvalues (relaxation rates) of the Uehling-Uhlenbeck collision operator have an upper cutoff. This cutoff requires summation over all discrete eigenvalues and eigenvectors of the collision operator when computing transport coefficients. We numerically compute the shear viscosity and thermal conductivity for any boson gas that interacts via a contact potential. We find that the bulk viscosity of the degenerate boson gas remains identically zero, as it is for the classical gas.
Riccati equation for simulation of leads in quantum transport
Bravi, M.; Farchioni, R.; Grosso, G.; Pastori Parravicini, G.
2014-10-01
We present a theoretical procedure with numerical demonstration of a workable and efficient method to evaluate the surface Green's function of semi-infinite leads connected to a device. Such a problem always occurs in quantum transport calculations but also in the study of surfaces and heterojunctions. We show here that these semi-infinite leads can be properly described by real-energy Green's functions obtained analytically by a smart solution of the Riccati matrix equation. The performance of our method is demonstrated in the case of a multichain two-dimensional electron-gas system, composed of a central ribbon connected to two semi-infinite leads, pierced by two opposite magnetic fields.
Application study of transport intensity equation in quantitative phase reconstruction
Song, Xiaojun; Cheng, Wei; Wei, Chunjuan; Xue, Liang; Liu, Weijing; Bai, Baodan; Chu, Fenghong
2016-10-01
In order to improve detection speed and accuracy of biological cells, a quantitative non-interference optical phase recovery method is proposed in commercial microscope, taking the red blood cells as the classical phase objects. Three bright field micrographs were collected in the experiment. Utilizing the transport intensity equation (TIE), the quantitative phase distributions of red blood cell are gained and agree well with the previous optical phase models. Analysis shows that the resolution of introduced system reaches sub-micron. This method not only quickly gives quantitative phase distribution of cells, but also measures a large number of cells simultaneously. So it is potential in the use of real-time observing and quantitative analyzing of cells in vivo.
Investigation of Resistivity of Saturated Porous Media with Lattice Boltzmann Method
Institute of Scientific and Technical Information of China (English)
YUE Wen-Zheng; TAO Guo; ZHU Ke-Qin
2004-01-01
The lattice Boltzmann method is employed to study the electrical transport properties of saturated porous media.Electrical current flow through the porous media is simulated and the relationship between resistivity index and water saturation is derived. It is found that this kind of relation is not a straight line as described by the Archie equation with the parameter n being a constant in a log-log scale. A new equation is thus developed to formulate this relation with n being a function of porosity and water saturation. The comparisons between the results by lattice Boltzmann and by the laboratory experiments on rock samples demonstrate that this numerical method can provide an alternative way for the expensive laboratory experiments to investigate the electrical transport properties of saturated porous media and can be used to explore micro mechanisms more conveniently.
Shafei, B.; Huber, C.; Parmigiani, A.; Taillefert, M.
2012-12-01
Physical and chemical heterogeneities associated with biogeochemical processes influence the fate and transport of contaminants in subsurface environments. We develop a new multi-species pore-scale reactive transport model based on the lattice Boltzmann method (LBM) to examine the temporal and spatial evolution of chemical species during the sorption of Arsenic. This model couples a fluid flow solver to an optimal advection-diffusion transport model where transport and reactions between chemical species are solved iteratively yielding a better stability and accuracy over a wide range of peclet numbers. It has already been applied to study 1) the permeability change of a porous medium during dissolution and precipitation and 2) the effect of spatial and chemical heterogeneities on the uptake of arsenic from the aqueous solution. By combining these two scenarios, we extend the model to incorporate arsenic speciation (i.e. As(III) and As(V)) and solid iron phase transformation, explore the distribution of iron, arsenic and partitioning of arsenic on various iron bearing solid phases. We investigate how the multitude of pore-domains affects the formation of redox gradients. As(III) and magnetite concentrations increase toward the anoxic zones while ferrihydrite and As(V) remains the dominant species in oxic conditions. The proposed reaction network includes: biotic reduction of ferrihydrite and magnetite to Fe2+(aq), of ferrihydrite to magnetite, biologically-mediated organic matter oxidation coupled with reduction of O2(aq) and As(V) , abiotic oxidation of Fe(II) by O2(aq) and sorption of As(V) and As(III) on Fe (hydr)oxide(s). All of these reactions are treated as kinetically controlled except As(V) and As(III) adsorption/desorption reactions which are expressed by equilibrium mass action laws. Similar set of reactions has been applied to simulate the distribution of As within constructed soil aggregates using continuum-scale model MIN3P (Masue-Slowey et al., 2010
Influence of asperities on fluid and thermal flow in a fracture: a coupled Lattice Boltzmann study
Neuville, Amélie; Toussaint, Renaud
2013-01-01
The characteristics of the hydro-thermal flow which occurs when a cold fluid is injected into a hot fractured bedrock depend on the morphology of the fracture. We consider a sharp triangular asperity, invariant in one direction, perturbing an otherwise flat fracture. We investigate its influence on the macroscopic hydraulic transmissivity and heat transfer efficiency, at fixed low Reynolds number. In this study, numerical simulations are done with a coupled lattice Boltzmann method that solves both the complete Navier-Stokes and advection-diffusion equations in three dimensions. The results are compared with those obtained under lubrication approximations which rely on many hypotheses and neglect the three-dimensional (3D) effects. The lubrication results are obtained by analytically solving the Stokes equation and a two-dimensional (integrated over the thickness) advection-diffusion equation. We use a lattice Boltzmann method with a double distribution (for mass and energy transport) on hypercubic and cubic ...
Probabilistic Analysis of the Upwind Scheme for Transport Equations
Delarue, François; Lagoutière, Frédéric
2011-01-01
We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we recover recent results due to Merlet and Vovelle (Numer Math 106: 129-155, 2007) and Merlet (SIAM J Numer Anal 46(1):124-150, 2007): we prove that the scheme is of order 1/2 in {L^{infty}([0,T],L^1(mathbb R^d))} for an integrable initial datum of bounded variation and of order 1/2- ɛ, for all ɛ > 0, in {L^{infty}([0,T] × mathbb R^d)} for an initial datum of Lipschitz regularity. Our analysis provides a new interpretation of the numerical diffusion phenomenon.
Forms of Approximate Radiation Transport
Brunner, G
2002-01-01
Photon radiation transport is described by the Boltzmann equation. Because this equation is difficult to solve, many different approximate forms have been implemented in computer codes. Several of the most common approximations are reviewed, and test problems illustrate the characteristics of each of the approximations. This document is designed as a tutorial so that code users can make an educated choice about which form of approximate radiation transport to use for their particular simulation.
Adaptive Lattice Boltzmann Model for Compressible Flows
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
A new lattice Boltzmann model for compressible flows is presented. The main difference from the standard lattice Boltzmann model is that the particle velocities are no longer constant, but vary with the mean velocity and internal energy. The adaptive nature of the particle velocities permits the mean flow to have a high Mach number. The introduction of a particle potential energy makes the model suitable for a perfect gas with arbitrary specific heat ratio. The Navier-Stokes (N-S) equations are derived by the Chapman-Enskog method from the BGK Boltzmann equation. Two kinds of simulations have been carried out on the hexagonal lattice to test the proposed model. One is the Sod shock-tube simulation. The other is a strong shock of Mach number 5.09 diffracting around a corner.
Elliptic random-walk equation for suspension and tracer transport in porous media
DEFF Research Database (Denmark)
Shapiro, Alexander; Bedrikovetsky, P. G.
2008-01-01
We propose a new approach to transport of the suspensions and tracers in porous media. The approach is based on a modified version of the continuous time random walk (CTRW) theory. In the framework of this theory we derive an elliptic transport equation. The new equation contains the time...
Spherical harmonics method for neutron transport equation based on unstructured-meshes
Institute of Scientific and Technical Information of China (English)
CAO Liang-Zhi; WU Hong-Chun
2004-01-01
Based on a new second-order neutron transport equation, self-adjoint angular flux (SAAF) equation, the spherical harmonics (PN) method for neutron transport equation on unstructured-meshes is derived. The spherical harmonics function is used to expand the angular flux. A set of differential equations about the spatial variable, which are coupled with each other, can be obtained. They are solved iteratively by using the finite element method on unstructured-meshes. A two-dimension transport calculation program is coded according to the model. The numerical results of some benchmark problems demonstrate that this method can give high precision results and avoid the ray effect very well.
Energy Technology Data Exchange (ETDEWEB)
Chatterjee, Kausik, E-mail: kausik.chatterjee@aggiemail.usu.edu [Strategic and Military Space Division, Space Dynamics Laboratory, North Logan, UT 84341 (United States); Center for Atmospheric and Space Sciences, Utah State University, Logan, UT 84322 (United States); Roadcap, John R., E-mail: john.roadcap@us.af.mil [Air Force Research Laboratory, Kirtland AFB, NM 87117 (United States); Singh, Surendra, E-mail: surendra-singh@utulsa.edu [Department of Electrical Engineering, The University of Tulsa, Tulsa, OK 74104 (United States)
2014-11-01
The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.
High Order Numerical Solution of Integral Transport Equation in Slab Geometry
Institute of Scientific and Technical Information of China (English)
沈智军; 袁光伟; 沈隆钧
2002-01-01
@@ There are some common numerical methods for solving neutron transport equation, which including the well-known discrete ordinates method, PN approximation and integral transport methods[1]. There exists certain singularities in the solution of transport equation near the boundary and interface[2]. It gives rise to the difficulty in the construction of high order accurate numerical methods. The numerical solution obtained by now can not attain the second order convergent accuracy[3,4].
DEFF Research Database (Denmark)
Lykke, Lars; Iversen, Bo Brummerstedt; Madsen, Georg
2006-01-01
The band structure of the low-temperature thermoelectric material, CsBi4Te6, is calculated and analyzed using the semiclassic transport equations. It is shown that to obtain a quantitative agreement with measured transport properties, a band gap of 0.08 eV must be enforced. A gap in reasonable...
A COMPARATIVE STUDY OF SOME OF THE SEDIMENT TRANSPORT EQUATIONS FOR AN ALLUVIAL CHANNEL WITH DUNES
Directory of Open Access Journals (Sweden)
Vajapeyam Srirangachar Srinivasan
2008-06-01
Full Text Available The present work is a comparative evaluation of some of the well known sediment transport equations for the condition of dunes on the bed. It is fairly clear that no single equation provides reliable estimates of the total load of sediment transported for all types of bed forms. The most frequently occurring bed form being dunes, only this case is considered in this paper. The measurements of sediment transport were realized in the laboratory for various sediment sizes, utilizing a computerized tilting recirculation flume. The Yang equation (1973 was found to provide the best results for dunes.
Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation
Directory of Open Access Journals (Sweden)
J. F. Gómez Aguilar
2016-01-01
Full Text Available The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range (0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to 1.
Boltzmann-Electron Model in Aleph.
Energy Technology Data Exchange (ETDEWEB)
Hughes, Thomas Patrick; Hooper, Russell
2014-11-01
We apply the Boltzmann-electron model in the electrostatic, particle-in-cell, finite- element code Aleph to a plasma sheath. By assuming a Boltzmann energy distribution for the electrons, the model eliminates the need to resolve the electron plasma fre- quency, and avoids the numerical "grid instability" that can cause unphysical heating of electrons. This allows much larger timesteps to be used than with kinetic electrons. Ions are treated with the standard PIC algorithm. The Boltzmann-electron model re- quires solution of a nonlinear Poisson equation, for which we use an iterative Newton solver (NOX) from the Trilinos Project. Results for the spatial variation of density and voltage in the plasma sheath agree well with an analytic model
Sun, Shuyu
2012-06-02
A new technique for the numerical solution of the partial differential equations governing transport phenomena in porous media is introduced. In this technique, the governing equations as depicted from the physics of the problem are used without extra manipulations. In other words, there is no need to reduce the number of governing equations by some sort of mathematical manipulations. This technique enables the separation of the physics part of the problem and the solver part, which makes coding more robust and could be used in several other applications with little or no modifications (e.g., multi-phase flow in porous media). In this method, one abandons the need to construct the coefficient matrix for the pressure equation. Alternatively, the coefficients are automatically generated within the solver routine. We show examples of using this technique to solving several flow problems in porous media.
Energy Technology Data Exchange (ETDEWEB)
Merton, S. R.; Smedley-Stevenson, R. P. [Computational Physics Group, AWE Aldermaston, Reading, Berkshire RG7 4PR (United Kingdom); Pain, C. C. [Dept. of Earth Science and Engineering, Imperial College London, London SW7 2AZ (United Kingdom)
2012-07-01
This paper describes a Non-Linear Discontinuous Petrov-Galerkin method and its application to the one-speed Boltzmann Transport Equation (BTE) for space-time problems. The purpose of the method is to remove unwanted oscillations in the transport solution which occur in the vicinity of sharp flux gradients, while improving computational efficiency and numerical accuracy. This is achieved by applying artificial dissipation in the solution gradient direction, internal to an element using a novel finite element (FE) Riemann approach. The added dissipation is calculated at each node of the finite element mesh based on local behaviour of the transport solution on both the spatial and temporal axes of the problem. Thus a different dissipation is used in different elements. The magnitude of dissipation that is used is obtained from a gradient-informed scaling of the advection velocities in the stabilisation term. This makes the method in its most general form non-linear. The method is implemented within a very general finite element Riemann framework. This makes it completely independent of choice of angular basis function allowing one to use different descriptions of the angular variation. Results show the non-linear scheme performs consistently well in demanding time-dependent multi-dimensional neutron transport problems. (authors)
Spherical-Harmonic Expansion Solution of Classical Transport Equations of Quark
Institute of Scientific and Technical Information of China (English)
CHEN Xiang-Jun; WANG Gang
2003-01-01
The spherical-harmonic method of solving classical transport equation of quark is investigated. Thehydrodynamics description of QGP as well as the relation between diffusion approximation and collective flow in nuclearcollisions are discussed.
Spherical-Harmonic Expansion Solution of Classical Transport Equations of Quark
Institute of Scientific and Technical Information of China (English)
CHENXiang-Jun; WANGGang
2003-01-01
The spherical-harmonic method of solving classical transport equation of quark is investigated. The hydrodynamics description of QGP as well as the relation between diffusion approximation and collective flow in nuclear collisions are discussed.
Gluon transport equations with condensate in the small angle approximation
Energy Technology Data Exchange (ETDEWEB)
Blaizot, Jean-Paul [Institut de Physique Théorique (IPhT), CNRS/URA2306, CEA Saclay, F-91191 Gif-sur-Yvette (France); Liao, Jinfeng [Physics Department and Center for Exploration of Energy and Matter, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States)
2016-05-15
We derive the set of kinetic equations that control the evolution of gluons in the presence of a condensate. We show that the dominant singularities remain logarithmic when the scattering involves particles in the condensate. This allows us to define a consistent small angle approximation.
Transition study of 3D aerodynamic configures using improved transport equations modeling
Institute of Scientific and Technical Information of China (English)
Xu Jiakuan; Bai Junqiang; Zhang Yang; Qiao Lei
2016-01-01
As boundary layer transition plays an important role in aerodynamic drag prediction, the proposal and study of transition prediction methods simulating the complex flow phenomena are prerequisite for aerodynamic design. In this paper, with the application of the linear stability theory based on amplification factor transport transition equations on the two-equation shear stress transport (SST) eddy-viscosity model, a new method, the SST-NTS-NCF model, is yielded. The new amplification factor transport equation for the crossflow instability induced transition is proposed to add to the NTS equation proposed by Coder, which simulates Tollmien–Schlichting wave tran-sition. The turbulent kinetic energy equation is modified by introducing a new source term that sim-ulates the transition process without the intermittency factor equation. Finally, coupled with these two amplification factor transport equations and SST turbulence model, a four-equation transition turbulence model is built. Comparisons between predictions using the new model and wind-tunnel experiments of NACA64(2)A015, NLF(2)-0415 and ONERA-D infinite swept wing and ONERA-M6 swept wing validate the predictive quality of the new SST-NTS-NCF model.
Transition study of 3D aerodynamic configures using improved transport equations modeling
Directory of Open Access Journals (Sweden)
Xu Jiakuan
2016-08-01
Full Text Available As boundary layer transition plays an important role in aerodynamic drag prediction, the proposal and study of transition prediction methods simulating the complex flow phenomena are prerequisite for aerodynamic design. In this paper, with the application of the linear stability theory based on amplification factor transport transition equations on the two-equation shear stress transport (SST eddy-viscosity model, a new method, the SST-NTS-NCF model, is yielded. The new amplification factor transport equation for the crossflow instability induced transition is proposed to add to the NTS equation proposed by Coder, which simulates Tollmien–Schlichting wave transition. The turbulent kinetic energy equation is modified by introducing a new source term that simulates the transition process without the intermittency factor equation. Finally, coupled with these two amplification factor transport equations and SST turbulence model, a four-equation transition turbulence model is built. Comparisons between predictions using the new model and wind-tunnel experiments of NACA64(2A015, NLF(2-0415 and ONERA-D infinite swept wing and ONERA-M6 swept wing validate the predictive quality of the new SST-NTS-NCF model.
Self-similar Solutions for a Transport Equation with Non-local Flux
Institute of Scientific and Technical Information of China (English)
Angel CASTRO; Diego C(O)RDOBA
2009-01-01
The authors construct self-similar solutions for an N-dimensional transport equation,where the velocity is given by the Riezs transform.These solutions imply nonuniqueness of weak solution.In addition,self-similar solution for a one-dimensional conservative equation involving the Hilbert transform is obtained.
Campos-García, Manuel; Granados-Agustín, Fermín.; Cornejo-Rodríguez, Alejandro; Estrada-Molina, Amilcar; Avendaño-Alejo, Maximino; Moreno-Oliva, Víctor Iván.
2013-11-01
In order to obtain a clearer interpretation of the Intensity Transport Equation (ITE), in this work, we propose an algorithm to solve it for some particular wavefronts and its corresponding intensity distributions. By simulating intensity distributions in some planes, the ITE is turns into a Poisson equation with Neumann boundary conditions. The Poisson equation is solved by means of the iterative algorithm SOR (Simultaneous Over-Relaxation).
Role of Dielectric Constant on Ion Transport: Reformulated Arrhenius Equation
Directory of Open Access Journals (Sweden)
Shujahadeen B. Aziz
2016-01-01
Full Text Available Solid and nanocomposite polymer electrolytes based on chitosan have been prepared by solution cast technique. The XRD results reveal the occurrence of complexation between chitosan (CS and the LiTf salt. The deconvolution of the diffractogram of nanocomposite solid polymer electrolytes demonstrates the increase of amorphous domain with increasing alumina content up to 4 wt.%. Further incorporation of alumina nanoparticles (6 to 10 wt.% Al2O3 results in crystallinity increase (large crystallite size. The morphological (SEM and EDX analysis well supported the XRD results. Similar trends of DC conductivity and dielectric constant with Al2O3 concentration were explained. The TEM images were used to explain the phenomena of space charge and blocking effects. The reformulated Arrhenius equation (σ(ε′,T=σoexp(-Ea/KBε′T was proposed from the smooth exponential behavior of DC conductivity versus dielectric constant at different temperatures. The more linear behavior of DC conductivity versus 1000/(ɛ′×T reveals the crucial role of dielectric constant in Arrhenius equation. The drawbacks of Arrhenius equation can be understood from the less linear behavior of DC conductivity versus 1000/T. The relaxation processes have been interpreted in terms of Argand plots.
The adjoint neutron transport equation and the statistical approach for its solution
Saracco, Paolo; Ravetto, Piero
2016-01-01
The adjoint equation was introduced in the early days of neutron transport and its solution, the neutron importance, has ben used for several applications in neutronics. The work presents at first a critical review of the adjoint neutron transport equation. Afterwards, the adjont model is constructed for a reference physical situation, for which an analytical approach is viable, i.e. an infinite homogeneous scattering medium. This problem leads to an equation that is the adjoint of the slowing-down equation that is well-known in nuclear reactor physics. A general closed-form analytical solution to such adjoint equation is obtained by a procedure that can be used also to derive the classical Placzek functions. This solution constitutes a benchmark for any statistical or numerical approach to the adjoint equation. A sampling technique to evaluate the adjoint flux for the transport equation is then proposed and physically interpreted as a transport model for pseudo-particles. This can be done by introducing appr...
A numerical method for the elliptic Monge-Amp\\`ere equation with transport boundary conditions
Froese, Brittany D
2011-01-01
The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\\`ere equation. While recent years have seen much work in the development of numerical methods for solving this equation, very little has been done on the implementation of the transport boundary conditions. In this paper, we propose a method for solving the transport problem by iteratively solving a Monge-Amp\\`ere equation with Neumann boundary conditions. We present a new discretization for the equation, which converges to the viscosity solution. The resulting system is solved efficiently with Newton's method. We provide several challenging computational examples that demonstrate the effectiveness and efficiency ($O(M)-O(M^{1.3})$ time) of the proposed method.
Least-squares finite element discretizations of neutron transport equations in 3 dimensions
Energy Technology Data Exchange (ETDEWEB)
Manteuffel, T.A [Univ. of Colorado, Boulder, CO (United States); Ressel, K.J. [Interdisciplinary Project Center for Supercomputing, Zurich (Switzerland); Starkes, G. [Universtaet Karlsruhe (Germany)
1996-12-31
The least-squares finite element framework to the neutron transport equation introduced in is based on the minimization of a least-squares functional applied to the properly scaled neutron transport equation. Here we report on some practical aspects of this approach for neutron transport calculations in three space dimensions. The systems of partial differential equations resulting from a P{sub 1} and P{sub 2} approximation of the angular dependence are derived. In the diffusive limit, the system is essentially a Poisson equation for zeroth moment and has a divergence structure for the set of moments of order 1. One of the key features of the least-squares approach is that it produces a posteriori error bounds. We report on the numerical results obtained for the minimum of the least-squares functional augmented by an additional boundary term using trilinear finite elements on a uniform tesselation into cubes.
Explicit solutions of the radiative transport equation in the P{sub 3} approximation
Energy Technology Data Exchange (ETDEWEB)
Liemert, André, E-mail: andre.liemert@ilm.uni-ulm.de; Kienle, Alwin [Institut für Lasertechnologien in der Medizin und Meßtechnik an der Universität Ulm, Helmholtzstr.12, Ulm D-89081 (Germany)
2014-11-01
Purpose: Explicit solutions of the monoenergetic radiative transport equation in the P{sub 3} approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light. Methods: A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P{sub 3} equations in closed form. Results: The derived solutions are illustrated and compared to exact solutions of the radiative transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P{sub 3} approximation is close to the exact solution of the radiative transport equation. Conclusions: The authors derived exact analytical solutions of the P{sub 3} equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium.
Barakat, A. R.; Schunk, R. W.
1982-01-01
A wide variety of plasma flow conditions is found in aeronomy and space plasma physics. Transport equations based on an isotropic Maxwellian vilecity distribution function can be used to describe plasma flows which contain 'small' temperature anisotropies. However, for plasma flows characterized by large temperature anisotropies, transport equations based on an anisotropic bi-Maxwellian (or two-temperature) velocity distribution function are expected to provide a much better description of the plasma transport properties. The present investigation is concerned with the extent to which transport equations based on both Maxwellian and bi-Maxwellian series expansions can describe plasma flows characterized by non-Maxwellian velocity distributions, giving particular attention to a modelling of the anisotropic character of the distribution function. The obtained results should provide clues as to the extent to which a given series expansion can account for the anisotropic character of a plasma.
Institute of Scientific and Technical Information of China (English)
2008-01-01
A discrete ordinates method for a threedimensional first-order neutron transport equation based on unstructured-meshes that avoids the singularity of the second-order neutron transport equation in void regions was derived.The finite element variation equation was obtained using the least-squares method.A three-dimensional transport calculation code was developed.Both the triangular-z and the tetrahedron elements were included.The numerical results of some benchmark problems demonstrated that this method can solve neutron transport problems in unstructuredmeshes very well.For most problems,the error of the eigenvalue and the angular flux is less than 0.3% and 3.0% respectively.
Pérez Guerrero, J. S.; Skaggs, T. H.
2010-08-01
SummaryMathematical models describing contaminant transport in heterogeneous porous media are often formulated as an advection-dispersion transport equation with distance-dependent transport coefficients. In this work, a general analytical solution is presented for the linear, one-dimensional advection-dispersion equation with distance-dependent coefficients. An integrating factor is employed to obtain a transport equation that has a self-adjoint differential operator, and a solution is found using the generalized integral transform technique (GITT). It is demonstrated that an analytical expression for the integrating factor exists for several transport equation formulations of practical importance in groundwater transport modeling. Unlike nearly all solutions available in the literature, the current solution is developed for a finite spatial domain. As an illustration, solutions for the particular case of a linearly increasing dispersivity are developed in detail and results are compared with solutions from the literature. Among other applications, the current analytical solution will be particularly useful for testing or benchmarking numerical transport codes because of the incorporation of a finite spatial domain.
Training Restricted Boltzmann Machines
DEFF Research Database (Denmark)
Fischer, Asja
Restricted Boltzmann machines (RBMs) are probabilistic graphical models that can also be interpreted as stochastic neural networks. Training RBMs is known to be challenging. Computing the likelihood of the model parameters or its gradient is in general computationally intensive. Thus, training...
Schinabeck, C.; Erpenbeck, A.; Härtle, R.; Thoss, M.
2016-11-01
Within the hierarchical quantum master equation (HQME) framework, an approach is presented, which allows a numerically exact description of nonequilibrium charge transport in nanosystems with strong electronic-vibrational coupling. The method is applied to a generic model of vibrationally coupled transport considering a broad spectrum of parameters ranging from the nonadiabatic to the adiabatic regime and including both resonant and off-resonant transport. We show that nonequilibrium effects are important in all these regimes. In particular, in the off-resonant transport regime, the inelastic cotunneling signal is analyzed for a vibrational mode in full nonequilibrium, revealing a complex interplay of different transport processes and deviations from the commonly used G0/2 rule of thumb. In addition, the HQME approach is used to benchmark approximate master equation and nonequilibrium Green's function methods.
Relativistic quantum transport coefficients for second-order viscous hydrodynamics
Florkowski, Wojciech; Maksymiuk, Ewa; Ryblewski, Radoslaw; Strickland, Michael
2015-01-01
We express the transport coefficients appearing in the second-order evolution equations for bulk viscous pressure and shear stress tensor using Bose-Einstein, Boltzmann, and Fermi-Dirac statistics for the equilibrium distribution function and Grad's 14-moment approximation as well as the method of Chapman-Enskog expansion for the non-equilibrium part. Specializing to the case of boost-invariant and transversally homogeneous longitudinal expansion of the viscous medium, we compare the results obtained using the above methods with those obtained from the exact solution of massive 0+1d Boltzmann equation in the relaxation-time approximation. We show that compared to the 14-moment approximation, the hydrodynamic transport coefficients obtained using the Chapman-Enskog method result in better agreement with the exact solution of the Boltzmann equation in relaxation-time approximation.
The suspended sediment transport equation and its near-bed sediment flux
Institute of Scientific and Technical Information of China (English)
LI RuiJie; LUO Feng; ZHU WenJin
2009-01-01
The suspended sediment transport equation and its near-bed sediment flux are one of the key prob-lems of sediment transport research under nonequilibrium condition. Based on the three-dimensional primitive suspended transport equation, the two-dimensional suspended sediment transport equation is deduced. The derived process indicates that the physical essence of the near-bed sediment flux is right the bottom boundary condition for the suspended sediment transport equation. This paper ana-lyzes the internal relations between the two methods of sediment carrying capacity and shear stress in common use, points out the consistency of these two methods in terms of form and physical meaning, and unifies these two methods theoretically. Furthermore, based on the analysis and comparison of the expressions of the near-bed sediment flux, this paper summarizes some problems to which attention should be paid, thus offering a novel approach to the study and the solution of the problems of sus-pended sediment transport and exchange flux of near-bed water sediment.
The suspended sediment transport equation and its near-bed sediment flux
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The suspended sediment transport equation and its near-bed sediment flux are one of the key problems of sediment transport research under nonequilibrium condition. Based on the three-dimensional primitive suspended transport equation, the two-dimensional suspended sediment transport equation is deduced. The derived process indicates that the physical essence of the near-bed sediment flux is right the bottom boundary condition for the suspended sediment transport equation. This paper analyzes the internal relations between the two methods of sediment carrying capacity and shear stress in common use, points out the consistency of these two methods in terms of form and physical meaning, and unifies these two methods theoretically. Furthermore, based on the analysis and comparison of the expressions of the near-bed sediment flux, this paper summarizes some problems to which attention should be paid, thus offering a novel approach to the study and the solution of the problems of suspended sediment transport and exchange flux of near-bed water sediment.
Lindner, Manfred
2007-01-01
Boltzmann equations are often used to describe the non-equilibrium time-evolution of many-body systems in particle physics. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process, which is described by so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the complete Kadanoff-Baym equations. Therefore, we present in this article a detailed comparison of Boltzmann and Kadanoff-Baym equations in the framework of a chirally invariant Yukawa-type quantum field theory including fermions and scalars. The obtained numerical results reveal significant differences between both types of equations. Apart from quantitative differences, on a qualitative level the late-time universality respected by Kadanoff-Baym equations is severely restricted in th...
A Unified Theory of Non-Ideal Gas Lattice Boltzmann Models
Luo, Li-Shi
1998-01-01
A non-ideal gas lattice Boltzmann model is directly derived, in an a priori fashion, from the Enskog equation for dense gases. The model is rigorously obtained by a systematic procedure to discretize the Enskog equation (in the presence of an external force) in both phase space and time. The lattice Boltzmann model derived here is thermodynamically consistent and is free of the defects which exist in previous lattice Boltzmann models for non-ideal gases. The existing lattice Boltzmann models for non-ideal gases are analyzed and compared with the model derived here.
A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers
Schüler, L.; Suciu, N.; Knabner, P.; Attinger, S.
2016-10-01
Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. The mixing model, describing the transport of the PDF in concentration space, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modelling.
Examination of transport equations pertaining to permeable elastic tubules such as Henle's loop.
Basmadjian, D; Baines, A D
1978-01-01
The transport equations applicable to loops of Henle and similar elastic permeable tubules were re-examined to assess the effect of radial transport resistance in the lumen and tubule geometry on solute transport. Active transport at the wall as well as external gradients equivalent to a 2--1,000-fold concentration increase per centimeter of distance were considered. Wall permeabilities and active transport constants were varied up to 2 . 10(-2) cm/s. It is shown that for conditions applicable to the loop of Henle, resistance to radial solute transfer in the lumen is negligible, both for passive and active transmural transport with concomitant water flux, and that axial dispersion further reduces that resistance. These conclusions apply equally to conical and elliptical geometries likely to arise in loop operation. The validity of Poiseuille's equation for these geometries is discussed. Ii is concluded that the one-dimensional transport equations are a valid representation of loop operation. Images FIGURE 1 PMID:737282
The H sub N method for solving linear transport equation: theory and applications
Tezcan, C; Guelecyuez, M C
2003-01-01
The system of singular integral equations which is obtained from the integro-differential form of the linear transport equation using the Placzek lemma is solved. The exit distributions at the boundaries of the various media and the infinite medium Green's function are used. The process is applied to the half-space and finite slab problems. The neutron angular density in terms of singular eigenfunctions of the method of elementary solutions is also used to derive the same analytical expressions.
Stability analysis of a system coupled to a transport equation using integral inequalities
Baudouin, Lucie; Seuret, Alexandre; Safi, Mohammed
2016-01-01
International audience; We address the stability of a system of ordinary differential equations coupled with a transport partial differential equation, using a Lyapunov functional approach. This system can also be interpreted as a finite dimensional system subject to a state delay. Inspired from recent developments on time-delay systems, a novel method to assess stability of such a class of coupled systems is developed here. We will specifically take advantage of a polynomial approximation of...
The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit
Energy Technology Data Exchange (ETDEWEB)
Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2016-07-12
We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radiation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material temperatures are always different in time-dependent problems, and we use it to show that IMC does not produce the correct diffusion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.
Solving the transport equation with quadratic finite elements: Theory and applications
Energy Technology Data Exchange (ETDEWEB)
Ferguson, J.M. [Lawrence Livermore National Lab., CA (United States)
1997-12-31
At the 4th Joint Conference on Computational Mathematics, the author presented a paper introducing a new quadratic finite element scheme (QFEM) for solving the transport equation. In the ensuing year the author has obtained considerable experience in the application of this method, including solution of eigenvalue problems, transmission problems, and solution of the adjoint form of the equation as well as the usual forward solution. He will present detailed results, and will also discuss other refinements of his transport codes, particularly for 3-dimensional problems on rectilinear and non-rectilinear grids.
Kinetic theory the Chapman-Enskog solution of the transport equation for moderately dense gases
Brush, S G
1972-01-01
Kinetic Theory, Volume 3: The Chapman-Enskog Solution of the Transport Equation for Moderately Dense Gases describes the Chapman-Enskog solution of the transport equation for moderately dense gases. Topics covered range from the propagation of sound in monatomic gases to the kinetic theory of simple and composite monatomic gases and generalizations of the theory to higher densities. The application of kinetic theory to the determination of intermolecular forces is also discussed. This volume is divided into two sections and begins with an introduction to the work of Hilbert, Chapman, and Ensko
h-Refinement for simple corner balance scheme of SN transport equation on distorted meshes
Yang, Rong; Yuan, Guangwei
2016-11-01
The transport sweep algorithm is a common method for solving discrete ordinate transport equation, but it breaks down once a concave cell appears in spatial meshes. To deal with this issue a local h-refinement for simple corner balance (SCB) scheme of SN transport equation on arbitrary quadrilateral meshes is presented in this paper by using a new subcell partition. It follows that a hybrid mesh with both triangle and quadrilateral cells is generated, and the geometric quality of these cells improves, especially it is ensured that all cells become convex. Combining with the original SCB scheme, an adaptive transfer algorithm based on the hybrid mesh is constructed. Numerical experiments are presented to verify the utility and accuracy of the new algorithm, especially for some application problems such as radiation transport coupled with Lagrangian hydrodynamic flow. The results show that it performs well on extremely distorted meshes with concave cells, on which the original SCB scheme does not work.
New Travelling Wave Solutions of Burgers Equation with Finite Transport Memory
Sakthivel, Rathinasamy; Chun, Changbum; Lee, Jonu
2010-09-01
The nonlinear evolution equations with finite memory have a wide range of applications in science and engineering. The Burgers equation with finite memory transport (time-delayed) describes convection-diffusion processes. In this paper, we establish the new solitary wave solutions for the time-delayed Burgers equation. The extended tanh method and the exp-function method have been employed to reveal these new solutions. Further, we have calculated the numerical solutions of the time-delayed Burgers equation with initial conditions by using the homotopy perturbation method (HPM). Our results show that the extended tanh and exp-function methods are very effective in finding exact solutions of the considered problem and HPM is very powerful in finding numerical solutions with good accuracy for nonlinear partial differential equations without any need of transformation or perturbation
Directory of Open Access Journals (Sweden)
Shulin Wu
2009-01-01
Full Text Available We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed-point equations with vector form. Based on the fixed-point vector equations, we introduce a new algorithm, namely, two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form.
Explicit finite-difference lattice Boltzmann method for curvilinear coordinates.
Guo, Zhaoli; Zhao, T S
2003-06-01
In this paper a finite-difference-based lattice Boltzmann method for curvilinear coordinates is proposed in order to improve the computational efficiency and numerical stability of a recent method [R. Mei and W. Shyy, J. Comput. Phys. 143, 426 (1998)] in which the collision term of the Boltzmann Bhatnagar-Gross-Krook equation for discrete velocities is treated implicitly. In the present method, the implicitness of the numerical scheme is removed by introducing a distribution function different from that being used currently. As a result, an explicit finite-difference lattice Boltzmann method for curvilinear coordinates is obtained. The scheme is applied to a two-dimensional Poiseuille flow, an unsteady Couette flow, a lid-driven cavity flow, and a steady flow around a circular cylinder. The numerical results are in good agreement with the results of previous studies. Extensions to other lattice Boltzmann models based on nonuniform meshes are also discussed.
Laboure, Vincent Matthieu
In this dissertation, we focus on solving the linear Boltzmann equation -- or transport equation -- using spherical harmonics (PN) expansions with fully-implicit time-integration schemes and Galerkin Finite Element spatial discretizations within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. The presentation is composed of two main ensembles. On one hand, we study the first-order form of the transport equation in the context of Thermal Radiation Transport (TRT). This nonlinear application physically necessitates to maintain a positive material temperature while the PN approximation tends to create oscillations and negativity in the solution. To mitigate these flaws, we provide a fully-implicit implementation of the Filtered PN (FPN) method and investigate local filtering strategies. After analyzing its effect on the conditioning of the system and showing that it improves the convergence properties of the iterative solver, we numerically investigate the error estimates derived in the linear setting and observe that they hold in the non-linear case. Then, we illustrate the benefits of the method on a standard test problem and compare it with implicit Monte Carlo (IMC) simulations. On the other hand, we focus on second-order forms of the transport equation for neutronics applications. We mostly consider the Self-Adjoint Angular Flux (SAAF) and Least-Squares (LS) formulations, the former being globally conservative but void incompatible and the latter having -- in all generality -- the opposite properties. We study the relationship between these two methods based on the weakly-imposed LS boundary conditions. Equivalences between various parity-based PN methods are also established, in particular showing that second-order filters are not an appropriate fix to retrieve void compatibility. The importance of global conservation is highlighted on a heterogeneous multigroup k-eigenvalue test problem. Based on these considerations, we propose a new
Jin, Jinshuang; Zheng, Xiao; Yan, YiJing
2008-06-21
A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.
A Derivation of the Nonlocal Volume-Averaged Equations for Two-Phase Flow Transport
Directory of Open Access Journals (Sweden)
Gilberto Espinosa-Paredes
2012-01-01
Full Text Available In this paper a detailed derivation of the general transport equations for two-phase systems using a method based on nonlocal volume averaging is presented. The local volume averaging equations are commonly applied in nuclear reactor system for optimal design and safe operation. Unfortunately, these equations are limited to length-scale restriction and according with the theory of the averaging volume method, these fail in transition of the flow patterns and boundaries between two-phase flow and solid, which produce rapid changes in the physical properties and void fraction. The non-local volume averaging equations derived in this work contain new terms related with non-local transport effects due to accumulation, convection diffusion and transport properties for two-phase flow; for instance, they can be applied in the boundary between a two-phase flow and a solid phase, or in the boundary of the transition region of two-phase flows where the local volume averaging equations fail.
Quantum transport in 1d systems via a master equation approach: numerics and an exact solution
Znidaric, Marko
2010-01-01
We discuss recent findings about properties of quantum nonequilibrium steady states. In particular we focus on transport properties. It is shown that the time dependent density matrix renormalization method can be used successfully to find a stationary solution of Lindblad master equation. Furthermore, for a specific model an exact solution is presented.
Stochastic approach to the numerical solution of the non-stationary Parker's transport equation
Wawrzynczak, A.; Modzelewska, R.; Gil, A.
2015-01-01
We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the turbulent medium is the Fokker-Planck type. It is the second order parabolic time-dependent 4-dimensional (3 spatial coordinates and particles energy/rigidity) partial differential equation. It is worth to mention that, if we assume the stationary case it remains as the 3-D parabolic type problem with respect to the particles rigidity R. If we fix the energy/rigidity it still remains as the 3-D parabolic type problem with respect to time. The proposed method of numerical solution is based on the solution of the system of stochastic differential equations (SDEs) being equivalent to the Parker's transport equation. We present the method of deriving from PTE the equivalent SDEs in the heliocentric spherical coordinate system for the backward approach. The advantages and disadvantages of the forward and the backward solution of the PTE are discussed. The obtained stochastic model of the Forbush decrease of the GCR intensity is in an agreement with the experimental data.
Bal, Guillaume; Schotland, John C
2015-01-01
We propose a method to reconstruct the density of an optical source in a highly scattering medium from ultrasound-modulated optical measurements. Our approach is based on the solution to a hybrid inverse source problem for the radiative transport equation (RTE). A controllability result for the RTE plays an essential role in the analysis.
Measuring the contour of a wavefront using the Irradiance Transport Equation (ITE)
Castillo-Rodríguez, Luis; Granados-Agustín, Fermín; Fernández-Guasti, Manuel; Cornejo-Rodríguez, Alejandro
2006-01-01
The Irradiance Transport Equation (ITE), found by Teague, had been used in optics with different applications. One of the field where had been used is in optical testing, for example, with the method developed by Takeda. In this paper following the idea of using different optical and mathematical analysis method, theorical and experimental results are presented.
Stochastic approach to the numerical solution of the non-stationary Parker's transport equation
Wawrzynczak, A; Gil, A
2015-01-01
We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the turbulent medium is the Fokker-Planck type. It is the second order parabolic time-dependent 4-dimensional (3 spatial coordinates and particles energy/rigidity) partial differential equation. It is worth to mention that, if we assume the stationary case it remains as the 3-D parabolic type problem with respect to the particles rigidity R. If we fix the energy it still remains as the 3-D parabolic type problem with respect to time. The proposed method of numerical solution is based on the solution of the system of stochastic differential equations (SDEs) being equivalent to the Parker's transport equation. We present the method of deriving from PTE the equivalent SDEs in the heliocentric spherical coordinate system for the backward approach. The obtained stochastic model of the Forbu...
Directory of Open Access Journals (Sweden)
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
Homogenization of the Poisson-Nernst-Planck Equations for Ion Transport in Charged Porous Media
Schmuck, Markus
2012-01-01
Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic ion transport in charged porous media. Homogenization analysis is performed for a two-component pe- riodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuous dielectric matrix, which is impermeable to the ions and carries a given surface charge. Three new features arise in the upscaled equations: (i) the effective ionic diffusivities and mobilities become tensors, related to the microstructure; (ii) the effective permittivity is also a tensor, depending on the electrolyte/matrix permittivity ratio and the ratio of the Debye screening length to mean pore size; and (iii) the surface charge per volume appears as a continuous "background charge density". The coeffcient tensors in the macroscopic PNP equations can be calculated from periodic reference cell problem, and several examples are considered. For an insulating solid matrix, all gradients are corrected by a single tortuosit...
Energy Technology Data Exchange (ETDEWEB)
Besnard, D. (Los Alamos National Lab., NM (United States) CEA Centre d' Etudes de Limeil, 94 - Villeneuve-Saint-Georges (France)); Harlow, F.H.; Rauenzahn, R.M.; Zemach, C. (Los Alamos National Lab., NM (United States))
1992-06-01
This study gives an updated account of our current ability to describe multimaterial compressible turbulent flows by means of a one-point transport model. Evolution equations are developed for a number of second-order correlations of turbulent data, and approximations of the gradient type are applied to additional correlations to close the system of equations. The principal fields of interest are the one- point Reynolds tensor for variable-density flow, the turbulent energy dissipation rate, and correlations for density-velocity and density- density fluctuations. This single-field description of turbulent flows is compared in some detail to two-field flow equations for nonturbulent, highly dispersed flow with separate variables for each field. This comparison suggests means for improved modeling of some correlations not subjected to evolution equations.
Noaman, B. A.; Korman, C. E.
2009-04-01
In this paper, we present a deterministic approach to calculate terminal current noise characteristics in semiconductor devices in the framework of semiclassical transport based on the spherical harmonics of the Boltzmann Transport Equation. The model relies on the solution of the Boltzmann equation in the frequency domain with special initial and boundary conditions. The terminal current fluctuation is directly related to scattering without the additional Langevin noise term added to the calculation. Simulation results are presented for the terminal current spectral density for a 1-D n+nn+ structure due to elastic-acoustic and intervally scattering.
Nonequilibrium phenomena in QCD and BEC. Boltzmann and beyond
Energy Technology Data Exchange (ETDEWEB)
Stockamp, T.
2006-12-22
In chapter 2 we chose the real time formalism to discuss some basic principles in quantum field theory at finite temperature. This enables us to derive the quantum Boltzmann equation from the Schwinger-Dyson series. We then shortly introduce the basic concepts of QCD which are needed to understand the physics of QGP formation. After a detailed account on the bottom-up scenario we show the consistency of this approach by a diagramatical analysis of the relevant Boltzmann collision integrals. Chapter 3 deals with BEC dynamics out of equilibrium. After an introduction to the fundamental theoretical tool - namely the Gross-Pitaevskii equation - we focus on a generalization to finite temperature developed by Zaremba, Nikuni and Griffin (ZNG). These authors use a Boltzmann equation to describe the interactions between condensed and excited atoms and manage in this way to describe condensate growth. We then turn to a discussion on the 2PI effective action and derive equations of motion for a relativistic scalar field theory. In the nonrelativistic limit these equations are shown to coincide with the ZNG theory when a quasiparticle approximation is applied. Finally, we perform a numerical analysis of the full 2PI equations. These remain valid even at strong coupling and far from equilibrium, and thus go far beyond Boltzmann's approach. For simplicity, we limit ourselves to a homogeneous system and present the first 3+1 dimensional study of condensate melting. (orig.)
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Morel, J.E.
1981-01-01
A collocation method is developed for the solution of the one-dimensional neutron transport equation in slab geometry with both symmetric and polarly asymmetric scattering. For the symmetric scattering case, it is found that the collocation method offers a combination of some of the best characteristics of the finite-element and discrete-ordinates methods. For the asymmetric scattering case, it is found that the computational cost of cross-section data processing under the collocation approach can be significantly less than that associated with the discrete-ordinates approach. A general diffusion equation treating both symmetric and asymmetric scattering is developed and used in a synthetic acceleration algorithm to accelerate the iterative convergence of collocation solutions. It is shown that a certain type of asymmetric scattering can radically alter the asymptotic behavior of the transport solution and is mathematically equivalent within the diffusion approximation to particle transport under the influence of an electric field. The method is easily extended to other geometries and higher dimensions. Applications exist in the areas of neutron transport with highly anisotropic scattering (such as that associated with hydrogenous media), charged-particle transport, and particle transport in controlled-fusion plasmas. 23 figures, 6 tables.
Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.
Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew
2014-12-26
Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.
Anderson-Witting transport coefficients for flows in general relativity
Ambrus, Victor E
2016-01-01
The transport coefficients induced by the Anderson-Witting approximation of the collision term in the relativistic Boltzmann equation are derived for close to equilibrium flows in general relativity. Using the tetrad formalism, it is shown that the expression for these coefficients is the same as that obtained on flat space-time, in agreement with the generalized equivalence principle.
Lattice Boltzmann method and its applications in engineering thermophysics
Institute of Scientific and Technical Information of China (English)
HE YaLing; LI Qing; WANG Yong; TANG GuiHua
2009-01-01
The lattice Boltzmann method (LBM),a mesoscopic method between the molecular dynamics method and the conventional numerical methods,has been developed into a very efficient numerical alternative in the past two decades.Unlike conventional numerical methods,the kinetic theory based LBM simulates fluid flows by tracking the evolution of the particle distribution function,and then accumulates the distribution to obtain macroscopic averaged properties.In this article we review some work on LBM applications in engineering thermophysics:(1) brief introduction to the development of the LBM; (2)fundamental theory of LBM including the Boltzmann equation,Maxwell distribution function,Boltzmann-BGK equation,and the lattice Boltzmann-BGK equation; (3) lattice Boltzmann models for compressible flows and non-equilibrium gas flows,bounce back-specular-reflection boundary scheme for microscale gaseous flows,the mass modified outlet boundary scheme for fully developed flows,and an implicit-explicit finite-difference-based LBM; and (4) applications of the LBM to oscillating flow,compressible flow,porous media flow,non-equilibrium flow,and gas resonant oscillating flow.
Note on the Solution of Transport Equation by Tau Method and Walsh Functions
Directory of Open Access Journals (Sweden)
Abdelouahab Kadem
2010-01-01
Full Text Available We consider the combined Walsh function for the three-dimensional case. A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions. This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. The main characteristic of this technique is that it reduces the problems to those of solving a system of algebraic equations; thus, it is greatly simplifying the problem.
Photons in a partonic transport approach
Energy Technology Data Exchange (ETDEWEB)
Greif, Moritz; Senzel, Florian; Greiner, Carsten [Goethe Universitaet Frankfurt, Max-von-Laue-Str. 1 60438 Frankfurt am Main (Germany)
2015-07-01
Partonic transport approaches have proved to be valuable tools in describing the quark-gluon plasma, created in heavy-ion collisions. In this work, first steps towards a dynamical understanding of photonproduction in expanding heavy-ion collisions are presented. Several photon production processes are included in the partonic cascade BAMPS (Boltzmann Approach to Multi-Parton Scatterings). BAMPS provides a microscopic tool to study expanding fireballs, employing a stochastic method to solve the relativistic 3+1d Boltzmann equation. Subsequently, photon spectra can be investigated, and in particular, the influence of the quark-gluon plasma phase for the elliptic flow of photons is studied.
Learning thermodynamics with Boltzmann machines
Torlai, Giacomo; Melko, Roger G.
2016-10-01
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.
Lattice Boltzmann Model for Compressible Fluid on a Square Lattice
Institute of Scientific and Technical Information of China (English)
SUN Cheng-Hai
2000-01-01
A two-level four-direction lattice Boltzmann model is formulated on a square lattice to simulate compressible flows with a high Mach number. The particle velocities are adaptive to the mean velocity and internal energy. Therefore, the mean flow can have a high Mach number. Due to the simple form of the equilibrium distribution, the 4th order velocity tensors are not involved in the calculations. Unlike the standard lattice Boltzmann model, o special treatment is need for the homogeneity of 4th order velocity tensors on square lattices. The Navier-Stokes equations were derived by the Chapman-Enskog method from the BGK Boltzmann equation. The model can be easily extended to three-dimensional cubic lattices. Two-dimensional shock-wave propagation was simulated
An integrable 3D lattice model with positive Boltzmann weights
Mangazeev, Vladimir V; Sergeev, Sergey M
2013-01-01
In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0Boltzmann weights.
Energy Technology Data Exchange (ETDEWEB)
Fevotte, F. [CEA Saclay, Dept. Modelisation de Systemes et Structures (DEN/DANS/DM2S/SERMA), 91 - Gif sur Yvette (France)
2008-07-01
At the various stages of a nuclear reactor's life, numerous studies are needed to guaranty the safety and efficiency of the design, analyse the fuel cycle, prepare the dismantlement, and so on. Due to the extreme difficulty to take extensive and accurate measurements in the reactor core, most of these studies are numerical simulations. The complete numerical simulation of a nuclear reactor involves many types of physics: neutronics, thermal hydraulics, materials, control engineering, Among these, the neutron transport simulation is one of the fundamental steps, since it allows computation - among other things - of various fundamental values such as the power density (used in thermal hydraulics computations) or fuel burn-up. The neutron transport simulation is based on the Boltzmann equation, which models the neutron population inside the reactor core. Among the various methods allowing its numerical solution, much interest has been devoted in the past few years to the Method of Characteristics in unstructured meshes (MOC), since it offers a good accuracy and operates in complicated geometries. The aim of this work is to propose improvements of the calculation scheme bound on the two dimensions MOC, in order to decrease the needed resources number. (A.L.B.)
Electron and ion transport equations in computational weakly-ionized plasmadynamics
Energy Technology Data Exchange (ETDEWEB)
Parent, Bernard [Department of Aerospace Engineering, Pusan National University, Busan 609-735 (Korea, Republic of); Macheret, Sergey O.; Shneider, Mikhail N. [Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544-5263 (United States)
2014-02-15
A new set of ion and electron transport equations is proposed to simulate steady or unsteady quasi-neutral or non-neutral multicomponent weakly-ionized plasmas through the drift–diffusion approximation. The proposed set of equations is advantaged over the conventional one by being considerably less stiff in quasi-neutral regions because it can be integrated in conjunction with a potential equation based on Ohm's law rather than Gauss's law. The present approach is advantaged over previous attempts at recasting the system by being applicable to plasmas with several types of positive ions and negative ions and by not requiring changes to the boundary conditions. Several test cases of plasmas enclosed by dielectrics and of glow discharges between electrodes show that the proposed equations yield the same solution as the standard equations but require 10 to 100 times fewer iterations to reach convergence whenever a quasi-neutral region forms. Further, several grid convergence studies indicate that the present approach exhibits a higher resolution (and hence requires fewer nodes to reach a given level of accuracy) when ambipolar diffusion is present. Because the proposed equations are not intrinsically linked to specific discretization or integration schemes and exhibit substantial advantages with no apparent disadvantage, they are generally recommended as a substitute to the fluid models in which the electric field is obtained from Gauss's law as long as the plasma remains weakly-ionized and unmagnetized.
Talamo, Alberto
2013-05-01
This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps.
Energy Technology Data Exchange (ETDEWEB)
Jin, Jinshuang, E-mail: jsjin@hznu.edu.cn [Department of Physics, Hangzhou Normal University, Hangzhou 310036 (China); Li, Jun [Department of Physics, Hangzhou Normal University, Hangzhou 310036 (China); College of Physics and Electronic Engineering, Dezhou University, Dezhou 253023 (China); Liu, Yu [State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083 (China); Li, Xin-Qi, E-mail: lixinqi@bnu.edu.cn [State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083 (China); Department of Physics, Beijing Normal University, Beijing 100875 (China); Department of Chemistry, Hong Kong University of Science and Technology, Kowloon (Hong Kong); Yan, YiJing, E-mail: yyan@ust.hk [Department of Chemistry, Hong Kong University of Science and Technology, Kowloon (Hong Kong); Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026 (China)
2014-06-28
Beyond the second-order Born approximation, we propose an improved master equation approach to quantum transport under self-consistent Born approximation. The basic idea is to replace the free Green's function in the tunneling self-energy diagram by an effective reduced propagator under the Born approximation. This simple modification has remarkable consequences. It not only recovers the exact results for quantum transport through noninteracting systems under arbitrary voltages, but also predicts the challenging nonequilibrium Kondo effect. Compared to the nonequilibrium Green's function technique that formulates the calculation of specific correlation functions, the master equation approach contains richer dynamical information to allow more efficient studies for such as the shot noise and full counting statistics.
Jin, Jinshuang; Li, Jun; Liu, Yu; Li, Xin-Qi; Yan, YiJing
2014-06-28
Beyond the second-order Born approximation, we propose an improved master equation approach to quantum transport under self-consistent Born approximation. The basic idea is to replace the free Green's function in the tunneling self-energy diagram by an effective reduced propagator under the Born approximation. This simple modification has remarkable consequences. It not only recovers the exact results for quantum transport through noninteracting systems under arbitrary voltages, but also predicts the challenging nonequilibrium Kondo effect. Compared to the nonequilibrium Green's function technique that formulates the calculation of specific correlation functions, the master equation approach contains richer dynamical information to allow more efficient studies for such as the shot noise and full counting statistics.
Modeling of the subgrid-scale term of the filtered magnetic field transport equation
Balarac, Guillaume; Kosovichev, Alexander; Brugière, Olivier; Wray, Alan; Mansour, Nagi
2010-01-01
Accurate subgrid-scale turbulence models are needed to perform realistic numerical magnetohydrodynamic (MHD) simulations of the subsurface flows of the Sun. To perform large-eddy simulations (LES) of turbulent MHD flows, three unknown terms have to be modeled. As a first step, this work proposes to use a priori tests to measure the accuracy of various models proposed to predict the SGS term appearing in the transport equation of the filtered magnetic field. It is proposed to evaluate the SGS ...
A stable scheme for computation of coupled transport and equilibrium equations in tokamaks
Fable, E.; Angioni, C.; Ivanov, A. A.; Lackner, K.; Maj, O.; Yu, S.; Medvedev; Pautasso, G.; Pereverzev, G. V.
2013-03-01
The coupled system consisting of 1D radial transport equations and the quasi-static 2D magnetic equilibrium equation for axisymmetric systems (tokamaks) is known to be prone to numerical instabilities, either due to propagation of numerical errors in the iteration process, or due to the choice of the numerical scheme itself. In this paper, a possible origin of these instabilities, specifically associated with the latter condition, is discussed and an approach is chosen, which is shown to have good accuracy and stability properties. This scheme is proposed to be used within those codes for which the poloidal flux ψ is the quantity solved for in the current diffusion equation. Mathematical arguments are used to study the convergence properties of the proposed scheme.
Directory of Open Access Journals (Sweden)
Peng Wei
2016-01-01
Full Text Available Tight schedules, multifunctional scopes, and colossal sizes usually characterize transportation megaprojects as challenging tasks for completion. In order to address these situations, a schedule risk management method was developed in this paper based on the structural equation model. In the proposed method, risk identification, evaluation and response were arranged as a sequence, and the expert elicitation technique was adopted in order to quantify the schedule risk status. To demonstrate the applicability of the proposed model, a megaproject case in China, the Shanghai Hongqiao Integrated Transport Hub (SHITH, was chosen. Information within the expanded risk register was collected including the probability and consequence of risk events, the complexity of risk responsible owners, the reaction time, and the time lasting for risk countermeasures. Final risk control results showed that the method could not only address the schedule risks correlations effectively, but also maintained the simplicity for construction management practices.
Quantum transport equations for low-dimensional multiband electronic systems: I.
Kupčić, I; Rukelj, Z; Barišić, S
2013-04-10
A systematic method of calculating the dynamical conductivity tensor in a general multiband electronic model with strong boson-mediated electron-electron interactions is described. The theory is based on the exact semiclassical expression for the coupling between valence electrons and electromagnetic fields and on the self-consistent Bethe-Salpeter equations for the electron-hole propagators. The general diagrammatic perturbation expressions for the intraband and interband single-particle conductivity are determined. The relations between the intraband Bethe-Salpeter equation, the quantum transport equation and the ordinary transport equation are briefly discussed within the memory-function approximation. The effects of the Lorentz dipole-dipole interactions on the dynamical conductivity of low-dimensional spα models are described in the same approximation. Such formalism proves useful in studies of different (pseudo)gapped states of quasi-one-dimensional systems with the metal-to-insulator phase transitions and can be easily extended to underdoped two-dimensional high-Tc superconductors.
Lattice-Boltzmann-based two-phase thermal model for simulating phase change.
Kamali, M R; Gillissen, J J J; van den Akker, H E A; Sundaresan, Sankaran
2013-09-01
A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A nonideal equation of state (EOS) is introduced by using a pseudopotential LB model. The evolution equation for the pseudotemperature variable is constructed in such a manner that in the continuum limit one recovers the well known macroscopic energy conservation equation for the mixtures. Heats of reaction, the enthalpy change associated with the phase change, and the diffusive transport of enthalpy are all taken into account; but the dependence of enthalpy on pressure, which is usually a small effect in most nonisothermal flows encountered in chemical reaction systems, is ignored. The energy equation is coupled to the LB equations for species transport and pseudopotential interaction forces through the EOS by using the filtered local pseudotemperature field. The proposed scheme is validated against simple test problems for which analytical solutions can readily be obtained.
Lattice Boltzmann Model for Numerical Relativity
Ilseven, E
2015-01-01
In the Bona-Masso formulation, Einstein equations are written as a set of flux conservative first order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for Numerical Relativity. Our model is validated with well-established tests, showing good agreement with analytical solutions. Furthermore, we show that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improves. Finally, in order to show the potential of our approach a linear scaling law for parallelisation with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.
Lattice Boltzmann model for numerical relativity.
Ilseven, E; Mendoza, M
2016-02-01
In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.
Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport
Energy Technology Data Exchange (ETDEWEB)
Litvinenko, Yuri E.; Effenberger, Frederic, E-mail: yuril@waikato.ac.nz [Department of Mathematics, University of Waikato, P.B. 3105 Hamilton (New Zealand)
2014-12-01
Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.
libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations
Jaruga, Anna; Jarecka, Dorota; Pawlowska, Hanna; Smolarkiewicz, Piotr K; Waruszewski, Maciej
2014-01-01
This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA). The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitati...
EXTERNAL BODY FORCE IN FINITE DIFFERENCE LATTICE BOLTZMANN METHOD
Institute of Scientific and Technical Information of China (English)
CHEN Sheng; LIU Zhao-hui; SHI Bao-chang; ZHENG Chu-guang
2005-01-01
A new finite difference lattice Boltzmann scheme is developed. Because of analyzing the influence of external body force roundly, the correct Navier-Stokes equations with the external body force are recovered, without any additional unphysical terms. And some numerical results are presented. The result which close agreement with analytical data shows the good performance of the model.
Institute of Scientific and Technical Information of China (English)
HUANG; Guanhua; HUANG; Quanzhong; ZHAN; Hongbin
2005-01-01
The newly developed Fractional Advection-Dispersion Equation (FADE), which is FADE was extended and used in this paper for modelling adsorbing contaminant transport by adding an adsorbing term. A parameter estimation method and its corresponding FORTRAN based program named FADEMain were developed on the basis of Nonlinear Least Square Algorithm and the analytical solution for one-dimensional FADE under the conditions of step input and steady state flow. Data sets of adsorbing contaminants Cd and NH4+-N transport in short homogeneous soil columns and conservative solute NaCI transport in a long homogeneous soil column, respectively were used to estimate the transport parameters both by FADEMain and the advection-dispersion equation (ADE) based program CXTFIT2.1. Results indicated that the concentration simulated by FADE agreed well with the measured data. Compared to the ADE model, FADE can provide better simulation for the concentration in the initial lower concentration part and the late higher concentration part of the breakthrough curves for both adsorbing contaminants. The dispersion coefficients for ADE were from 0.13 to 7.06 cm2/min, while the dispersion coefficients for FADE ranged from 0.119 to 3.05 cm1.856/min for NaCI transport in the long homogeneous soil column. We found that the dispersion coefficient of FADE increased with the transport distance, and the relationship between them can be quantified with an exponential function. Less scale-dependent was also found for the dispersion coefficient of FADE with respect to ADE.
Hu, Kainan; Geng, Shaojuan
2016-01-01
A new lattice Boltzmann scheme associated with flexible specific heat ratio is proposed. The new free degree is introduced via the internal energy associated with the internal structure. The evolution equation of the distribution function is reduced to two evolution equations. One is connected to the density and velocity, the other is of the energy. A two-dimensional lattice Boltzmann model and a three-dimensional lattice Boltzmann model are derived via the Hermite expansion. The two lattice Boltzmann models are applied to simulating the shock tube of one dimension. Good agreement between the numerical results and the analytical solutions are obtained.
Numerical solution of transport equation for applications in environmental hydraulics and hydrology
Rashidul Islam, M.; Hanif Chaudhry, M.
1997-04-01
The advective term in the one-dimensional transport equation, when numerically discretized, produces artificial diffusion. To minimize such artificial diffusion, which vanishes only for Courant number equal to unity, transport owing to advection has been modeled separately. The numerical solution of the advection equation for a Gaussian initial distribution is well established; however, large oscillations are observed when applied to an initial distribution with sleep gradients, such as trapezoidal distribution of a constituent or propagation of mass from a continuous input. In this study, the application of seven finite-difference schemes and one polynomial interpolation scheme is investigated to solve the transport equation for both Gaussian and non-Gaussian (trapezoidal) initial distributions. The results obtained from the numerical schemes are compared with the exact solutions. A constant advective velocity is assumed throughout the transport process. For a Gaussian distribution initial condition, all eight schemes give excellent results, except the Lax scheme which is diffusive. In application to the trapezoidal initial distribution, explicit finite-difference schemes prove to be superior to implicit finite-difference schemes because the latter produce large numerical oscillations near the steep gradients. The Warming-Kutler-Lomax (WKL) explicit scheme is found to be better among this group. The Hermite polynomial interpolation scheme yields the best result for a trapezoidal distribution among all eight schemes investigated. The second-order accurate schemes are sufficiently accurate for most practical problems, but the solution of unusual problems (concentration with steep gradient) requires the application of higher-order (e.g. third- and fourth-order) accurate schemes.
Application of lattice Boltzmann scheme to nanofluids
Institute of Scientific and Technical Information of China (English)
XUAN Yimin; LI Qiang; YAO Zhengping
2004-01-01
A nanofluid is a particle suspension that consists of base liquids and nanoparticles. Nanofluid has greater potential for heat transfer enhancement than traditional solid-liquid mixture. By accounting for the external and internal forces acting on the suspended nanoparticles and interactions among the nanoparticles and fluid particles,a lattice Boltzmann model for simulating flow and energy transport processes inside the nanofluids is proposed. The irregular motion of the nanoparticles and inherent dynamic behavior of nanofluids are discussed. The distributions of suspended nanoparticles inside nanofluids are calculated.
Energy Technology Data Exchange (ETDEWEB)
Humphrey, E. [Carnegie Mellon University, Pittsburgh, PA 15213 (United States); Phatak, C.; Petford-Long, A.K. [Argonne National Laboratory, Argonne, IL 60439 (United States); De Graef, M. [Carnegie Mellon University, Pittsburgh, PA 15213 (United States)
2014-04-01
We introduce a new approach for the separation of the electrostatic and magnetic components of the electron wave phase shift, based on the transport-of-intensity equation (TIE) formalism. We derive two separate TIE-like equations, one for each of the phase shift components. We use experimental results on FeCoB and Permalloy patterned islands to illustrate how the magnetic and electrostatic longitudinal derivatives can be computed. The main advantage of this new approach is the fact that the differences in the power spectra of the two phase components (electrostatic phase shifts often have significant power in the higher frequencies) can be accommodated by the selection of two different Tikhonov regularization parameters for the two phase reconstructions. The extra computational demands of the method are more than compensated by the improved phase reconstruction results. - Highlights: • We provide a new way to separate electrostatic and magnetic phase shifts in Lorentz microscopy. • We derive two new transport-of-intensity style equations, one for electrostatic phase shifts and the other for magnetic phase shifts. • We provide a new way to determine the longitudinal intensity derivative that automatically includes time reversal symmetry. • This approach allows for the Tikhonov regularization parameter to be selected for each phase shift separately. • We provide two example application on Permalloy and CoFeB patterned islands.
Contact line dynamics in binary lattice Boltzmann simulations
Pooley, C M; Yeomans, J M; 10.1103/PhysRevE.78.056709
2008-01-01
We show that, when a single relaxation time lattice Boltzmann algorithm is used to solve the hydrodynamic equations of a binary fluid for which the two components have different viscosities, strong spurious velocities in the steady state lead to incorrect results for the equilibrium contact angle. We identify the origins of these spurious currents, and demonstrate how the results can be greatly improved by using a lattice Boltzmann method based on a multiple-relaxation-time algorithm. By considering capillary filling we describe the dependence of the advancing contact angle on the interface velocity.
Usang, M. D.; Ivanyuk, F. A.; Ishizuka, C.; Chiba, S.
2016-10-01
Nuclear fission is treated by using the Langevin dynamical description with macroscopic and microscopic transport coefficients (mass and friction tensors), and it is elucidated how the microscopic (shell and pairing) effects in the transport coefficients, especially their dependence on temperature, affects various fission observables. We found that the microscopic transport coefficients, calculated by linear response theory, change drastically as a function of temperature: in general, the friction increases with growing temperature while the mass tensor decreases. This temperature dependence brings a noticeable change in the mass distribution and kinetic energies of fission fragments from nuclei around 236U at an excitation energy of 20 MeV. The prescission kinetic energy decreases from 25 MeV at low temperature to about 2.5 MeV at high temperature. In contrast, the Coulomb kinetic energy increases as the temperature increases. Interpolating the microscopic transport coefficients among the various temperatures enabled our Langevin equation to use the microscopic transport coefficients at a deformation-dependent local temperature of the dynamical evolution. This allowed us to compare directly the fission observables of both macroscopic and microscopic calculations, and we found almost identical results under the conditions considered in this work.
Kotake, Kei; Yamada, Shoichi; Takiwaki, Tomoya; Kuroda, Takami; Suwa, Yudai; Nagakura, Hiroki
2012-01-01
This is a status report on our endeavor to reveal the mechanism of core-collapse supernovae (CCSNe) by large-scale numerical simulations. Multi-dimensionality of the supernova engine, general relativisitic magnetohydrodynamics, energy and lepton number transport by neutrinos emitted from the forming neutron star as well as nuclear interactions there, are all believed to play crucial roles in repelling infalling matter and producing energetic explosions. These ingredients are nonlinearly coupled with one another in the dynamics of core-collapse, bounce, and shock expansion. Serious quantitative studies of CCSNe hence make extensive numerical computations mandatory. Since neutrinos are neither in thermal nor in chemical equilibrium in general, their distributions in the phase space should be computed. This is a six dimensional (6D) neutrino transport problem and quite a challenge even for those with an access to the most advanced numerical resources such as the "K computer". To tackle this problem, we have emba...
Directory of Open Access Journals (Sweden)
Thereza A. Soares
2004-08-01
Full Text Available The ability of biomolecules to catalyze chemical reactions is due chiefly to their sensitivity to variations of the pH in the surrounding environment. The reason for this is that they are made up of chemical groups whose ionization states are modulated by pH changes that are of the order of 0.4 units. The determination of the protonation states of such chemical groups as a function of conformation of the biomolecule and the pH of the environment can be useful in the elucidation of important biological processes from enzymatic catalysis to protein folding and molecular recognition. In the past 15 years, the theory of Poisson-Boltzmann has been successfully used to estimate the pKa of ionizable sites in proteins yielding results, which may differ by 0.1 unit from the experimental values. In this study, we review the theory of Poisson-Boltzmann under the perspective of its application to the calculation of pKa in proteins.
Lattice Boltzmann method for linear oscillatory noncontinuum flows
Shi, Yong; Yap, Ying Wan; Sader, John E.
2014-03-01
Oscillatory gas flows are commonly generated by micro- and nanoelectromechanical systems. Due to their small size and high operating frequencies, these devices often produce noncontinuum gas flows. Theoretical analysis of such flows requires solution of the unsteady Boltzmann equation, which can present a formidable challenge. In this article, we explore the applicability of the lattice Boltzmann (LB) method to such linear oscillatory noncontinuum flows; this method is derived from the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation. We formulate four linearized LB models in the frequency domain, based on Gaussian-Hermite quadratures of different algebraic precision (AP). The performance of each model is assessed by comparison to high-accuracy numerical solutions to the linearized Boltzmann-BGK equation for oscillatory Couette flow. The numerical results demonstrate that high even-order LB models provide superior performance over the greatest noncontinuum range. Our results also highlight intrinsic deficiencies in the current LB framework, which is incapable of capturing noncontinuum behavior at high oscillation frequencies, regardless of quadrature AP and the Knudsen number.
Nauenberg, Michael
2005-03-01
In 1916 Einstein published a remarkable paper entitled ``On the Quantum Theory of Radiation''ootnotetextA. Einstein ``On the Quantum theory of Radiation,'' Phys. Zeitschrift 18 (1917) 121. First printed in Mitteilungender Physikalischen Gesellschaft Zurich. No 18, 1916. Translated into English in Van der Waerden ``Sources of Quantum Mechanics'' (North Holland 1967) pp. 63-77. in which he obtained Planck's formula for black-body radiation by introducing a new statistical hypothesis for the emmision and absorption of electromagneic radiation based on discrete bundles of energy and momentum which are now called photons. Einstein radiation theory replaced Maxwell's classical theory by a stochastic process which, when properly interpreted, also gives well known statistics of massless particles with even spin.^2 This quantum distribution, however, was not discovered by Einstein but was communicated to him by Bose in 1924. Like Boltzmann's classical counterpart, Einstein's statistical theory leads to an irreversible approach to thermal equilibrium, but because this violates time reversal, Einstein theory can not be regarded as a fundamental theory of physical process.ootnotetextM. Nauenberg ``The evolution of radiation towards thermal equilibrium: A soluble model which illustrates the foundations of statistical mechanics,'' American Journal of Physics 72 (2004) 313 Apparently Einstein and his contemporaries were unaware of this problem, and even today this problem is ignored in contemporary discussions of Einstein's treatment of the black-body spectrum.
Bernal García, Álvaro; Abarca Giménez, Agustín; Barrachina Celda, Teresa María; Miró Herrero, Rafael
2014-01-01
This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of Computer Mathematics in 2014, available online: http://www.tandfonline.com/10.1080/00207160.2013.799668 Resolution of the steady-state Neutron Transport Equation in a nuclear pool reactor is usually achieved by means of two different numerical methods: Monte Carlo (stochastic) and Discrete Ordinates (deterministic). The Discrete Ordinates method solves the Neutron Transport Equation for a...
Numerical modeling of photon migration in human neck based on the radiative transport equation
Fujii, Hiroyuki; Nadamoto, Ken; Okada, Eiji; Yamada, Yukio; Hoshi, Yoko; Watanabe, Masao
2016-01-01
Biomedical optical imaging has a possibility of a comprehensive diagnosis of thyroid cancer in conjunction with ultrasound imaging. For improvement of the optical imaging, this study develops a higher order scheme for solving the time-dependent radiative transport equation (RTE) by use of the finite-difference and discrete-ordinate methods. The accuracy and efficiency of the developed scheme are examined by comparison with the analytical solutions of the RTE in homogeneous media. Then, the developed scheme is applied to describing photon migration in the human neck model. The numerical simulations show complex behaviors of photon migration in the human neck model due to multiple diffusive reflection near the trachea.
Gluon Transport Equation with Effective Mass and Dynamical Onset of Bose-Einstein Condensation
Blaizot, Jean-Paul; Liao, Jinfeng
2015-01-01
We study the transport equation describing a dense system of gluons, in the small scattering angle approximation, taking into account medium-generated effective masses of the gluons. We focus on the case of overpopulated systems that are driven to Bose-Einstein condensation on their way to thermalization. The presence of a mass modifies the dispersion relation of the gluon, as compared to the massless case, but it is shown that this does not change qualitatively the scaling behavior in the vicinity of the onset.
The Slab Albedo Problem Using Singular Eigenfunctions and the Third Form of the Transport Equation
Kaskas, Ayþe; Tezcan, Cevdet
1997-01-01
The albedo and the transmission factor for slabs are obtained using the infinite medium Green's function in terms of the singular eigenfunctions in the third form of the transport equation. Our analytical results are simple as in FN-method and the convergence of the numerical results is as faster as in the CN-method. Calculations are also carried out by various incoming angular fluxes and uncollided neutrons are taken into account. Our numerical results are in very good agreement with the results of the CN method.
Solution of transport equations in layered media with refractive index mismatch using the PN-method.
Phillips, Kevin G; Jacques, Steven L
2009-10-01
The PN-method is a spectral discretization technique used to obtain numerical solutions to the radiative transport equation. To the best of our knowledge, the PN-method has yet to be generalized to the case of refractive index mismatch in layered slabs used to numerically simulate skin. Our main contribution is the application of a collocation method that takes into account refractive index mismatch at layer interfaces. The stability, convergence, and accuracy of the method are established. Example calculations demonstrating the flexibility of the method are performed.
Energy Technology Data Exchange (ETDEWEB)
Bailey, T S; Chang, J H; Warsa, J S; Adams, M L
2010-12-22
We present a new spatial discretization of the discrete-ordinates transport equation in two-dimensional Cartesian (X-Y) geometry for arbitrary polygonal meshes. The discretization is a discontinuous finite element method (DFEM) that utilizes piecewise bi-linear (PWBL) basis functions, which are formally introduced in this paper. We also present a series of numerical results on quadrilateral and polygonal grids and compare these results to a variety of other spatial discretizations that have been shown to be successful on these grid types. Finally, we note that the properties of the PWBL basis functions are such that the leading-order piecewise bi-linear discontinuous finite element (PWBLD) solution will satisfy a reasonably accurate diffusion discretization in the thick diffusion limit, making the PWBLD method a viable candidate for many different classes of transport problems.
Finite-element discretization of 3D energy-transport equations for semiconductors
Energy Technology Data Exchange (ETDEWEB)
Gadau, Stephan
2007-07-01
In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and
Sengers, J. V.; Basu, R. S.; Sengers, J. M. H. L.
1981-01-01
A survey is presented of representative equations for various thermophysical properties of fluids in the critical region. Representative equations for the transport properties are included. Semi-empirical modifications of the theoretically predicted asymtotic critical behavior that yield simple and practical representations of the fluid properties in the critical region are emphasized.
Energy Technology Data Exchange (ETDEWEB)
Pinchedez, K
1999-06-01
Parallel computing meets the ever-increasing requirements for neutronic computer code speed and accuracy. In this work, two different approaches have been considered. We first parallelized the sequential algorithm used by the neutronics code CRONOS developed at the French Atomic Energy Commission. The algorithm computes the dominant eigenvalue associated with PN simplified transport equations by a mixed finite element method. Several parallel algorithms have been developed on distributed memory machines. The performances of the parallel algorithms have been studied experimentally by implementation on a T3D Cray and theoretically by complexity models. A comparison of various parallel algorithms has confirmed the chosen implementations. We next applied a domain sub-division technique to the two-group diffusion Eigen problem. In the modal synthesis-based method, the global spectrum is determined from the partial spectra associated with sub-domains. Then the Eigen problem is expanded on a family composed, on the one hand, from eigenfunctions associated with the sub-domains and, on the other hand, from functions corresponding to the contribution from the interface between the sub-domains. For a 2-D homogeneous core, this modal method has been validated and its accuracy has been measured. (author)
Lattice Boltzmann modelling of intrinsic permeability
Li, Jun; Wu, Lei; Zhang, Yonghao
2016-01-01
Lattice Boltzmann method (LBM) has been applied to predict flow properties of porous media including intrinsic permeability, where it is implicitly assumed that the LBM is equivalent to the incompressible (or near incompressible) Navier-Stokes equation. However, in LBM simulations, high-order moments, which are completely neglected in the Navier-Stokes equation, are still available through particle distribution functions. To ensure that the LBM simulation is correctly working at the Navier-Stokes hydrodynamic level, the high-order moments have to be negligible. This requires that the Knudsen number (Kn) is small so that rarefaction effect can be ignored. In this technical note, we elaborate this issue in LBM modelling of porous media flows, which is particularly important for gas flows in ultra-tight media.
Neutron transport with anisotropic scattering: theory and applications
Van den Eynde, Gert
2005-01-01
This thesis is a blend of neutron transport theory and numerical analysis. We start with the study of the problem of the Mika/Case eigenexpansion used in the solution process of the homogeneous one-speed Boltzmann neutron transport equation with anisotropic scattering for plane symmetry. The anisotropic scattering is expressed as a finite Legendre series in which the coefficients are the ``scattering coefficients'. This eigenexpansion consists of a discrete spectrum of eigenvalues with its co...
A numerical spectral approach to solve the dislocation density transport equation
Djaka, K. S.; Taupin, V.; Berbenni, S.; Fressengeas, C.
2015-09-01
A numerical spectral approach is developed to solve in a fast, stable and accurate fashion, the quasi-linear hyperbolic transport equation governing the spatio-temporal evolution of the dislocation density tensor in the mechanics of dislocation fields. The approach relies on using the Fast Fourier Transform algorithm. Low-pass spectral filters are employed to control both the high frequency Gibbs oscillations inherent to the Fourier method and the fast-growing numerical instabilities resulting from the hyperbolic nature of the transport equation. The numerical scheme is validated by comparison with an exact solution in the 1D case corresponding to dislocation dipole annihilation. The expansion and annihilation of dislocation loops in 2D and 3D settings are also produced and compared with finite element approximations. The spectral solutions are shown to be stable, more accurate for low Courant numbers and much less computation time-consuming than the finite element technique based on an explicit Galerkin-least squares scheme.
Simulation Study of the Electron and Hole Transport in a CNTFET
Institute of Scientific and Technical Information of China (English)
A.Bahari; M.Amiri
2013-01-01
In this work we have investigated electron and hole transport through zig zag carbon nanotubes by solving Boltzmann Transport Equation (BTE).We find that the mobility of electrons is rather greater than holes.Carbo nanotubes with longer diameter can carry higher current.Normally,transport of electrons (or holes) is dominated by scattering events,which relax the carrier momentum in an effort to bring the conducting material to equilibrium.
Quantum Simulator for Transport Phenomena in Fluid Flows
Mezzacapo, A; Lamata, L; Egusquiza, I L; Succi, S; Solano, E
2015-01-01
Transport phenomena are one of the most challenging problems in computational physics. We present a quantum simulator based on pseudospin-boson quantum systems, which is suitable for encoding fluid dynamics problems within a lattice kinetic formalism. This quantum simulator is obtained by exploiting the analogies between Dirac and lattice Boltzmann equations. It is shown that both the streaming and collision processes of lattice Boltzmann dynamics can be implemented with controlled quantum operations, using a heralded quantum protocol to encode non-unitary scattering processes. The proposed simulator is amenable to realization in controlled quantum platforms, such as ion-trap quantum computers or circuit quantum electrodynamics processors.
Gaeuman, D.; Andrews, E.D.; Kraus, A.; Smith, W.
2009-01-01
Bed load samples from four locations in the Trinity River of northern California are analyzed to evaluate the performance of the Wilcock-Crowe bed load transport equations for predicting fractional bed load transport rates. Bed surface particles become smaller and the fraction of sand on the bed increases with distance downstream from Lewiston Dam. The dimensionless reference shear stress for the mean bed particle size (t*rm) is largest near the dam, but varies relatively little between the more downstream locations. The relation between t*rm and the reference shear stresses for other size fractions is constant across all locations. Total bed load transport rates predicted with the Wilcock-Crowe equations are within a factor of 2 of sampled transport rates for 68% of all samples. The Wilcock-Crowe equations nonetheless consistently under-predict the transport of particles larger than 128 mm, frequently by more than an order of magnitude. Accurate prediction of the transport rates of the largest particles is important for models in which the evolution of the surface grain size distribution determines subsequent bed load transport rates. Values of term estimated from bed load samples are up to 50% larger than those predicted with the Wilcock-Crowe equations, and sampled bed load transport approximates equal mobility across a wider range of grain sizes than is implied by the equations. Modifications to theWilcock-Crowe equation for determining t*rm and the hiding function used to scale term to other grain size fractions are proposed to achieve the best fit to observed bed load transport in the Trinity River. Copyright 2009 by the American eophysical Union.
Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation
Energy Technology Data Exchange (ETDEWEB)
Chang, B; Brown, P; Greenbaum, A; Machorro, E
2005-01-21
In this paper, we prove that the numerical solution of the mono-directional neutron transport equation by the Petrov-Galerkin method converges to the true solution in the L{sup 2} norm at the rate of h{sup 2}. Since consistency has been shown elsewhere, the focus here is on stability. We prove that the system of Petrov-Galerkin equations is stable by showing that the 2-norm of the inverse of the matrix for the system of equations is bounded by a number that is independent of the order of the matrix. This bound is equal to the length of the longest path that it takes a neutron to cross the domain in a straight line. A consequence of this bound is that the global error of the Petrov-Galerkin approximation is of the same order of h as the local truncation error. We use this result to explain the widely held observation that the solution of the Petrov-Galerkin method is second accurate for one class of problems, but is only first order accurate for another class of problems.
Lattice Boltzmann simulations of multiple-droplet interaction dynamics
Zhou, Wenchao; Loney, Drew; Fedorov, Andrei G.; Degertekin, F. Levent; Rosen, David W.
2014-03-01
A lattice Boltzmann (LB) formulation, which is consistent with the phase-field model for two-phase incompressible fluid, is proposed to model the interface dynamics of droplet impingement. The interparticle force is derived by comparing the macroscopic transport equations recovered from LB equations with the governing equations of the continuous phase-field model. The inconsistency between the existing LB implementations and the phase-field model in calculating the relaxation time at the phase interface is identified and an approximation is proposed to ensure the consistency with the phase-field model. It is also shown that the commonly used equilibrium velocity boundary for the binary fluid LB scheme does not conserve momentum at the wall boundary and a modified scheme is developed to ensure the momentum conservation at the boundary. In addition, a geometric formulation of the wetting boundary condition is proposed to replace the popular surface energy formulation and results show that the geometric approach enforces the prescribed contact angle better than the surface energy formulation in both static and dynamic wetting. The proposed LB formulation is applied to simulating droplet impingement dynamics in three dimensions and results are compared to those obtained with the continuous phase-field model, the LB simulations reported in the literature, and experimental data from the literature. The results show that the proposed LB simulation approach yields not only a significant speed improvement over the phase-field model in simulating droplet impingement dynamics on a submillimeter length scale, but also better accuracy than both the phase-field model and the previously reported LB techniques when compared to experimental data. Upon validation, the proposed LB modeling methodology is applied to the study of multiple-droplet impingement and interactions in three dimensions, which demonstrates its powerful capability of simulating extremely complex interface
On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient
Directory of Open Access Journals (Sweden)
John D. Towers
2002-10-01
Full Text Available We study the Cauchy problem for the nonlinear (possibly strongly degenerate parabolic transport-diffusion equation $$ partial_t u + partial_x (gamma(xf(u=partial_x^2 A(u, quad A'(cdotge 0, $$ where the coefficient $gamma(x$ is possibly discontinuous and $f(u$ is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as $varepsilondownarrow 0$ in a suitable sequence ${u_{varepsilon}}_{varepsilon>0}$ of smooth approximations solving the problem above with the transport flux $gamma(xf(cdot$ replaced by $gamma_{varepsilon}(xf(cdot$ and the diffusion function $A(cdot$ replaced by $A_{varepsilon}(cdot$, where $gamma_{varepsilon}(cdot$ is smooth and $A_{varepsilon}'(cdot>0$. The main technical challenge is to deal with the fact that the total variation $|u_{varepsilon}|_{BV}$ cannot be bounded uniformly in $varepsilon$, and hence one cannot derive directly strong convergence of ${u_{varepsilon}}_{varepsilon>0}$. In the purely hyperbolic case ($A'equiv 0$, where existence has already been established by a number of authors, all existence results to date have used a singular maolinebreak{}pping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term. Submitted April 29, 2002. Published October 27, 2002. Math Subject Classifications: 35K65, 35D05, 35R05, 35L80 Key Words: Degenerate parabolic equation; nonconvex flux; weak solution; discontinuous coefficient; viscosity method; a priori estimates; compensated compactness
Directory of Open Access Journals (Sweden)
Pawlasova Pavlina
2015-12-01
Full Text Available Satisfaction is one of the key factors which influences customer loyalty. We assume that the satisfied customer will be willing to use the ssame service provider again. The overall passengers´ satisfaction with public city transport may be affected by the overall service quality. Frequency, punctuality, cleanliness in the vehicle, proximity, speed, fare, accessibility and safety of transport, information and other factors can influence passengers´ satisfaction. The aim of this paper is to quantify factors and identify the most important factors influencing customer satisfaction with public city transport within conditions of the Czech Republic. Two methods of analysis are applied in order to fulfil the aim. The method of factor analysis and the method Varimax were used in order to categorize variables according to their mutual relations. The method of structural equation modelling was used to evaluate the factors and validate the model. Then, the optimal model was found. The logistic parameters, including service continuity and frequency, and service, including information rate, station proximity and vehicle cleanliness, are the factors influencing passengers´ satisfaction on a large scale.
Equation of state dependence of directed flow in a microscopic transport model
Nara, Yasushi; Steinheimer, Jan; Stoecker, Horst
2016-01-01
We study the sensitivities of the directed flow in Au+Au collisions on the equation of state (EoS), employing the transport theoretical model JAM. The EoS is modified by introducing a new collision term in order to control the pressure of a system by appropriately selecting an azimuthal angle in two-body collisions according to a given EoS. It is shown that this approach is an efficient method to modify the EoS in a transport model. The beam energy dependence of the directed flow of protons is examined with two different EoS, a first-order phase transition and crossover. It is found that our approach yields quite similar results as hydrodynamical predictions on the beam energy dependence of the directed flow; Transport theory predicts a minimum in the excitation function of the slope of proton directed flow and does indeed yield negative directed flow, if the EoS with a first-order phase transition is employed. Our result strongly suggests that the highest sensitivity for the critical point can be seen in the...
Nomura, Yasunori
2015-01-01
Understanding the observed arrow of time is equivalent, under general assumptions, to explaining why Boltzmann brains do not overwhelm ordinary observers. It is usually thought that this provides a condition on the decay rate of every cosmologically accessible de Sitter vacuum, and that this condition is determined by the production rate of Boltzmann brains calculated using semiclassical theory built on each such vacuum. We argue, based on a recently developed picture of microscopic quantum gravitational degrees of freedom, that this thinking needs to be modified. In particular, depending on the structure of the fundamental theory, the decay rate of a de Sitter vacuum may not have to satisfy any condition except possibly the one imposed by the Poincare recurrence. The framework discussed here also addresses the question of whether a Minkowski vacuum may produce Boltzmann brains.
Lattice-Boltzmann simulation of laser interaction with weakly ionized helium plasmas.
Li, Huayu; Ki, Hyungson
2010-07-01
This paper presents a lattice Boltzmann method for laser interaction with weakly ionized plasmas considering electron impact ionization and three-body recombination. To simulate with physical properties of plasmas, the authors' previous work on the rescaling of variables is employed and the electromagnetic fields are calculated from the Maxwell equations by using the finite-difference time-domain method. To calculate temperature fields, energy equations are derived separately from the Boltzmann equations. In this way, we attempt to solve the full governing equations for plasma dynamics. With the developed model, the continuous-wave CO2 laser interaction with helium is simulated successfully.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Directory of Open Access Journals (Sweden)
Yoshinobu Tanaka
2012-01-01
Full Text Available The overall membrane pair characteristics included in the overall mass transport equation are understandable using the phenomenological equations expressed in the irreversible thermodynamics. In this investigation, the overall membrane pair characteristics (overall transport number , overall solute permeability , overall electro-osmotic permeability and overall hydraulic permeability were measured by seawater electrodialysis changing current density, temperature and salt concentration, and it was found that occasionally takes minus value. For understanding the above phenomenon, new concept of the overall concentration reflection coefficient ∗ is introduced from the phenomenological equation. This is the aim of this investigation. ∗ is defined for describing the permselectivity between solutes and water molecules in the electrodialysis system just after an electric current interruption. ∗ is expressed by the function of and . ∗ is generally larger than 1 and is positive, but occasionally ∗ becomes less than 1 and becomes negative. Negative means that ions are transferred with water molecules (solvent from desalting cells toward concentrating cells just after an electric current interruption, indicating up-hill transport or coupled transport between water molecules and solutes.
libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations
Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.
2015-04-01
This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA) on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations
Directory of Open Access Journals (Sweden)
A. Jaruga
2015-04-01
Full Text Available This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case; and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
Radiation Transport Calculations and Simulations
Energy Technology Data Exchange (ETDEWEB)
Fasso, Alberto; /SLAC; Ferrari, A.; /CERN
2011-06-30
This article is an introduction to the Monte Carlo method as used in particle transport. After a description at an elementary level of the mathematical basis of the method, the Boltzmann equation and its physical meaning are presented, followed by Monte Carlo integration and random sampling, and by a general description of the main aspects and components of a typical Monte Carlo particle transport code. In particular, the most common biasing techniques are described, as well as the concepts of estimator and detector. After a discussion of the different types of errors, the issue of Quality Assurance is briefly considered.
Hypocoercivity for linear kinetic equations conserving mass
Dolbeault, Jean
2015-02-03
We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf
Silicon wafer microstructure imaging using InfraRed Transport of Intensity Equation
Li, Hongru; Feng, Guoying; Bourgade, Thomas; Zuo, Chao; Du, Yongzhao; Zhou, Shouhuan; Asundi, Anand
2015-03-01
A novel quantitative 3D imaging system of silicon microstructures using InfraRed Transport of Intensity Equation (IRTIE) is proposed in this paper. By recording the intensity at multiple planes and using FFT or DCT based TIE solver, fast and accurate phase retrieval for both uniform and non-uniform intensity distributions is proposed. Numerical simulation and experiments confirm the accuracy and reliability of the proposed method. The application of IR-TIE for inspection of micro-patterns in visibly opaque media using 1310 nm light source is demonstrated. For comparison, micro-patterns are also inspected by the contact scanning mode Taylor Hobson system. Quantitative agreement suggests the possibility of using IR-TIE for phase imaging of silicon wafers.
A non-equilibrium equation-of-motion approach to quantum transport utilizing projection operators.
Ochoa, Maicol A; Galperin, Michael; Ratner, Mark A
2014-11-12
We consider a projection operator approach to the non-equilibrium Green function equation-of-motion (PO-NEGF EOM) method. The technique resolves problems of arbitrariness in truncation of an infinite chain of EOMs and prevents violation of symmetry relations resulting from the truncation (equivalence of left- and right-sided EOMs is shown and symmetry with respect to interchange of Fermi or Bose operators before truncation is preserved). The approach, originally developed by Tserkovnikov (1999 Theor. Math. Phys. 118 85) for equilibrium systems, is reformulated to be applicable to time-dependent non-equilibrium situations. We derive a canonical form of EOMs, thus explicitly demonstrating a proper result for the non-equilibrium atomic limit in junction problems. A simple practical scheme applicable to quantum transport simulations is formulated. We perform numerical simulations within simple models and compare results of the approach to other techniques and (where available) also to exact results.
A Transport Equation Approach to Green Functions and Self-force Calculations
Wardell, Barry
2010-01-01
In a recent work, we presented the first application of the Poisson-Wiseman-Anderson method of `matched expansions' to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the Green function which are respectively valid in the `quasilocal' and `distant past' regimes, and which may be matched together within the normal neighbourhood. In this article, we introduce the method of matched expansions and discuss transport equation methods for the calculation of the Green function in the quasilocal region. These methods allow the Green function to be evaluated throughout the normal neighborhood and are also relevant to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity.
Phase retrieval based on cosine grating modulation and transport of intensity equation
Chen, Ya-ping; Zhang, Quan-bing; Cheng, Hong; Qian, Yi; Lv, Qian-qian
2016-10-01
In order to calculate the lost phase from the intensity information effectively, a new method of phase retrieval which based on cosine grating modulation and transport of intensity equation is proposed. Firstly, the cosine grating is loaded on the spatial light modulator in the horizontal and vertical direction respectively, and the corresponding amplitude of the light field is modulated. Then the phase is calculated by its gradient which is extracted from different direction modulation light illumination. The capability of phase recovery of the proposed method in the presence of noise is tested by simulation experiments. And the results show that the proposed algorithm has a better resilience than the traditional Fourier transform algorithm at low frequency noise. Furthermore, the phase object of different scales can be retrieved using the proposed algorithm effectively by changing the frequency of cosine grating, which can control the imaging motion expediently.
Immiscible multicomponent lattice Boltzmann model for fluids with high relaxation time ratio
Indian Academy of Sciences (India)
Tao Jiang; Qiwei Gong; Ruofan Qiu; Anlin Wang
2014-10-01
An immiscible multicomponent lattice Boltzmann model is developed for fluids with high relaxation time ratios, which is based on the model proposed by Shan and Chen (SC). In the SC model, an interaction potential between particles is incorporated into the discrete lattice Boltzmann equation through the equilibrium velocity. Compared to the SC model, external forces in our model are discretized directly into the discrete lattice Boltzmann equation, as proposed by Guo et al. We develop it into a new multicomponent lattice Boltzmann (LB) model which has the ability to simulate immiscible multicomponent fluids with relaxation time ratio as large as 29.0 and to reduce `spurious velocity’. In this work, the improved model is validated and studied using the central bubble case and the rising bubble case. It finds good applications in both static and dynamic cases for multicomponent simulations with different relaxation time ratios.
Meng, Xin; Huang, Huachuan; Yan, Keding; Tian, Xiaolin; Yu, Wei; Cui, Haoyang; Kong, Yan; Xue, Liang; Liu, Cheng; Wang, Shouyu
2016-12-20
In order to realize high contrast imaging with portable devices for potential mobile healthcare, we demonstrate a hand-held smartphone based quantitative phase microscope using the transport of intensity equation method. With a cost-effective illumination source and compact microscope system, multi-focal images of samples can be captured by the smartphone's camera via manual focusing. Phase retrieval is performed using a self-developed Android application, which calculates sample phases from multi-plane intensities via solving the Poisson equation. We test the portable microscope using a random phase plate with known phases, and to further demonstrate its performance, a red blood cell smear, a Pap smear and monocot root and broad bean epidermis sections are also successfully imaged. Considering its advantages as an accurate, high-contrast, cost-effective and field-portable device, the smartphone based hand-held quantitative phase microscope is a promising tool which can be adopted in the future in remote healthcare and medical diagnosis.
Tian, Xiaolin; Meng, Xin; Yu, Wei; Song, Xiaojun; Xue, Liang; Liu, Cheng; Wang, Shouyu
2016-10-01
Microscopy combined with the transport of intensity equation is capable of retrieving both intensity and phase distributions of samples from both in-focus and defocus intensities. However, during measurements, the focal plane is often decided artificially and the improper choice may induce errors in quantitative intensity and phase retrieval. In order to obtain accurate in-focus information, quantitative intensity and phase imaging with the numerical focusing transport of intensity equation method combined with cellular duty ratio criterion and numerical wavefront propagation is introduced in this paper. Both numerical simulations and experimental measurements are provided proving this designed method can increase both retrieved in-focus intensity and phase accuracy and reduce dependence of focal plane determination in transport of intensity equation measurements. It is believed that the proposed method can be potentially applied in various fields as in-focus compensation for quantitative phase imaging and automatic focal plane determination, etc.
Egami, Yoshiyuki; Iwase, Shigeru; Tsukamoto, Shigeru; Ono, Tomoya; Hirose, Kikuji
2015-09-01
We develop a first-principles electron-transport simulator based on the Lippmann-Schwinger (LS) equation within the framework of the real-space finite-difference scheme. In our fully real-space-based LS (grid LS) method, the ratio expression technique for the scattering wave functions and the Green's function elements of the reference system is employed to avoid numerical collapse. Furthermore, we present analytical expressions and/or prominent calculation procedures for the retarded Green's function, which are utilized in the grid LS approach. In order to demonstrate the performance of the grid LS method, we simulate the electron-transport properties of the semiconductor-oxide interfaces sandwiched between semi-infinite jellium electrodes. The results confirm that the leakage current through the (001)Si-SiO_{2} model becomes much larger when the dangling-bond state is induced by a defect in the oxygen layer, while that through the (001)Ge-GeO_{2} model is insensitive to the dangling bond state.
A lattice Boltzmann model for adsorption breakthrough
Energy Technology Data Exchange (ETDEWEB)
Agarwal, Saurabh; Verma, Nishith [Indian Institute of Technology Kanpur, Department of Chemical Engineering, Kanpur (India); Mewes, Dieter [Universitat Hannover, Institut fur Verfahrenstechnik, Hannover (Germany)
2005-07-01
A lattice Boltzmann model is developed to simulate the one-dimensional (1D) unsteady state concentration profiles, including breakthrough curves, in a fixed tubular bed of non-porous adsorbent particles. The lattice model solves the 1D time dependent convection-diffusion-reaction equation for an ideal binary gaseous mixture, with solute concentrations at parts per million levels. The model developed in this study is also able to explain the experimental adsorption/desorption data of organic vapours (toluene) on silica gel under varying conditions of temperature, concentrations and flowrates. Additionally, the programming code written for simulating the adsorption breakthrough is modified with minimum changes to successfully simulate a few flow problems, such as Poiseuille flow, Couette flow, and axial dispersion in a tube. The present study provides an alternative numerical approach to solving such types of mass transfer related problems. (orig.)
LATTICE-BOLTZMANN MODEL FOR COMPRESSIBLE PERFECT GASES
Institute of Scientific and Technical Information of China (English)
Sun Chenghai
2000-01-01
We present an adaptive lattice Boltzmann model to simulate super sonic flows. The particle velocities are determined by the mean velocity and internal energy. The adaptive nature of particle velocities permits the mean flow to have high Mach number. A particle potential energy is introduced so that the model is suitable for the perfect gas with arbitrary specific heat ratio. The Navier-Stokes equations are derived by the Chapman-Enskog method from the BGK Boltzmann equation.As preliminary tests, two kinds of simulations have been performed on hexagonal lattices. One is the one-dimensional simulation for sinusoidal velocity distributions.The velocity distributions are compared with the analytical solution and the mea sured viscosity is compared with the theoretical values. The agreements are basically good. However, the discretion error may cause some non-isotropic effects. The other simulation is the 29 degree shock reflection.
Lattice Boltzmann model for incompressible flows through porous media.
Guo, Zhaoli; Zhao, T S
2002-09-01
In this paper a lattice Boltzmann model is proposed for isothermal incompressible flow in porous media. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium (the Darcy's term and the Forcheimer's term). Through the Chapman-Enskog procedure, the generalized Navier-Stokes equations for incompressible flow in porous media are derived from the present lattice Boltzmann model. The generalized two-dimensional Poiseuille flow, Couette flow, and lid-driven cavity flow are simulated using the present model. It is found the numerical results agree well with the analytical and/or the finite-difference solutions.
Energy Technology Data Exchange (ETDEWEB)
Azmy, Yousry
2014-06-10
We employ the Integral Transport Matrix Method (ITMM) as the kernel of new parallel solution methods for the discrete ordinates approximation of the within-group neutron transport equation. The ITMM abandons the repetitive mesh sweeps of the traditional source iterations (SI) scheme in favor of constructing stored operators that account for the direct coupling factors among all the cells' fluxes and between the cells' and boundary surfaces' fluxes. The main goals of this work are to develop the algorithms that construct these operators and employ them in the solution process, determine the most suitable way to parallelize the entire procedure, and evaluate the behavior and parallel performance of the developed methods with increasing number of processes, P. The fastest observed parallel solution method, Parallel Gauss-Seidel (PGS), was used in a weak scaling comparison with the PARTISN transport code, which uses the source iteration (SI) scheme parallelized with the Koch-baker-Alcouffe (KBA) method. Compared to the state-of-the-art SI-KBA with diffusion synthetic acceleration (DSA), this new method- even without acceleration/preconditioning-is completitive for optically thick problems as P is increased to the tens of thousands range. For the most optically thick cells tested, PGS reduced execution time by an approximate factor of three for problems with more than 130 million computational cells on P = 32,768. Moreover, the SI-DSA execution times's trend rises generally more steeply with increasing P than the PGS trend. Furthermore, the PGS method outperforms SI for the periodic heterogeneous layers (PHL) configuration problems. The PGS method outperforms SI and SI-DSA on as few as P = 16 for PHL problems and reduces execution time by a factor of ten or more for all problems considered with more than 2 million computational cells on P = 4.096.
Energy Technology Data Exchange (ETDEWEB)
Carella, Alfredo Raul
2012-09-15
Quantifying species transport rates is a main concern in chemical and petrochemical industries. In particular, the design and operation of many large-scale industrial chemical processes is as much dependent on diffusion as it is on reaction rates. However, the existing diffusion models sometimes fail to predict experimentally observed behaviors and their accuracy is usually insufficient for process optimization purposes. Fractional diffusion models offer multiple possibilities for generalizing Flick's law in a consistent manner in order to account for history dependence and nonlocal effects. These models have not been extensively applied to the study of real systems, mainly due to their computational cost and mathematical complexity. A least squares spectral formulation was developed for solving fractional differential equations. The proposed method was proven particularly well-suited for dealing with the numerical difficulties inherent to fractional differential operators. The practical implementation was explained in detail in order to enhance reproducibility, and directions were specified for extending it to multiple dimensions and arbitrarily shaped domains. A numerical framework based on the least-squares spectral element method was developed for studying and comparing anomalous diffusion models in pellets. This simulation tool is capable of solving arbitrary integro-differential equations and can be effortlessly adapted to various problems in any number of dimensions. Simulations of the flow around a cylindrical particle were achieved by extending the functionality of the developed framework. A test case was analyzed by coupling the boundary condition yielded by the fluid model with two families of anomalous diffusion models: hyperbolic diffusion and fractional diffusion. Qualitative guidelines for determining the suitability of diffusion models can be formulated by complementing experimental data with the results obtained from this approach.(Author)
A fast, parallel algorithm to solve the basic fluvial erosion/transport equations
Braun, J.
2012-04-01
Quantitative models of landform evolution are commonly based on the solution of a set of equations representing the processes of fluvial erosion, transport and deposition, which leads to predict the geometry of a river channel network and its evolution through time. The river network is often regarded as the backbone of any surface processes model (SPM) that might include other physical processes acting at a range of spatial and temporal scales along hill slopes. The basic laws of fluvial erosion requires the computation of local (slope) and non-local (drainage area) quantities at every point of a given landscape, a computationally expensive operation which limits the resolution of most SPMs. I present here an algorithm to compute the various components required in the parameterization of fluvial erosion (and transport) and thus solve the basic fluvial geomorphic equation, that is very efficient because it is O(n) (the number of required arithmetic operations is linearly proportional to the number of nodes defining the landscape), and is fully parallelizable (the computation cost decreases in a direct inverse proportion to the number of processors used to solve the problem). The algorithm is ideally suited for use on latest multi-core processors. Using this new technique, geomorphic problems can be solved at an unprecedented resolution (typically of the order of 10,000 X 10,000 nodes) while keeping the computational cost reasonable (order 1 sec per time step). Furthermore, I will show that the algorithm is applicable to any regular or irregular representation of the landform, and is such that the temporal evolution of the landform can be discretized by a fully implicit time-marching algorithm, making it unconditionally stable. I will demonstrate that such an efficient algorithm is ideally suited to produce a fully predictive SPM that links observationally based parameterizations of small-scale processes to the evolution of large-scale features of the landscapes on
High order fluid model for streamer discharges: I. Derivation of model and transport data
Dujko, S; White, R D; Ebert, U
2013-01-01
Streamer discharges pose basic problems in plasma physics, as they are very transient, far from equilibrium and have high ionization density gradients; they appear in diverse areas of science and technology. The present paper focuses on the derivation of a high order fluid model for streamers. Using momentum transfer theory, the fluid equations are obtained as velocity moments of the Boltzmann equation; they are closed in the local mean energy approximation and coupled to the Poisson equation for the space charge generated electric field. The high order tensor in the energy flux equation is approximated by the product of two lower order moments to close the system. The average collision frequencies for momentum and energy transfer in elastic and inelastic collisions for electrons in molecular nitrogen are calculated from a multi term Boltzmann equation solution. We then discuss, in particular, (1) the correct implementation of transport data in streamer models; (2) the accuracy of the two term approximation f...
Institute of Scientific and Technical Information of China (English)
Liming WU; Zhengliang ZHANG
2006-01-01
We establish Talagrand's T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type's approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to ReactionDiffusion equations are provided.
Indian Academy of Sciences (India)
Marko Žnidarič
2011-11-01
We discuss recent ﬁndings about properties of quantum nonequilibrium steady states. In particular we focus on transport properties. It is shown that the time-dependent density matrix renormalization method can be used successfully to ﬁnd a stationary solution of Lindblad master equation. Furthermore, for a speciﬁc model an exact solution is presented.
An integrated Boltzmann+hydrodynamics approach to heavy ion collisions
Energy Technology Data Exchange (ETDEWEB)
Petersen, Hannah
2009-04-22
In this thesis the first fully integrated Boltzmann+hydrodynamics approach to relativistic heavy ion reactions has been developed. After a short introduction that motivates the study of heavy ion reactions as the tool to get insights about the QCD phase diagram, the most important theoretical approaches to describe the system are reviewed. The hadron-string transport approach that this work is based on is the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach. Predictions for the charged particle multiplicities at LHC energies are made. The next step is the development of a new framework to calculate the baryon number density in a transport approach. Time evolutions of the net baryon number and the quark density have been calculated at AGS, SPS and RHIC energies. Studies of phase diagram trajectories using hydrodynamics are performed. The hybrid approach that has been developed as the main part of this thesis is based on the UrQMD transport approach with an intermediate hydrodynamical evolution for the hot and dense stage of the collision. The full (3+1) dimensional ideal relativistic one fluid dynamics evolution is solved using the SHASTA algorithm. Three different equations of state have been used, namely a hadron gas equation of state without a QGP phase transition, a chiral EoS and a bag model EoS including a strong first order phase transition. For the freeze-out transition from hydrodynamics to the cascade calculation two different set-ups are employed. The parameter dependences of the model are investigated and the time evolution of different quantities is explored. The hybrid model calculation is able to reproduce the experimentally measured integrated as well as transverse momentum dependent v{sub 2} values for charged particles. The multiplicity and mean transverse mass excitation function is calculated for pions, protons and kaons in the energy range from E{sub lab}=2-160 A GeV. The HBT correlation of the negatively charged pion source
Energy Technology Data Exchange (ETDEWEB)
Shadid, J.N.; Tuminaro, R.S. [Sandia National Labs., Albuquerque, NM (United States); Walker, H.F. [Utah State Univ., Logan, UT (United States). Dept. of Mathematics and Statistics
1997-02-01
The solution of the governing steady transport equations for momentum, heat and mass transfer in flowing fluids can be very difficult. These difficulties arise from the nonlinear, coupled, nonsymmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this manuscript the authors focus on evaluating a proposed nonlinear solution method based on an inexact Newton method with backtracking. In this context they use a particular spatial discretization based on a pressure stabilized Petrov-Galerkin finite element formulation of the low Mach number Navier-Stokes equations with heat and mass transport. The discussion considers computational efficiency, robustness and some implementation issues related to the proposed nonlinear solution scheme. Computational results are presented for several challenging CFD benchmark problems as well as two large scale 3D flow simulations.
Liu, Chongxuan; Szecsody, Jim E.; Zachara, John M.; Ball, William P.
The generalized integral transform technique (GITT) is applied to solve the one-dimensional advection-dispersion equation (ADE) in heterogeneous porous media coupled with either linear or nonlinear sorption and decay. When both sorption and decay are linear, analytical solutions are obtained using the GITT for one-dimensional ADEs with spatially and temporally variable flow and dispersion coefficient and arbitrary initial and boundary conditions. When either sorption or decay is nonlinear the solutions to ADEs with the GITT are hybrid analytical-numerical. In both linear and nonlinear cases, the forward and inverse integral transforms for the problems described in the paper are apparent and straightforward. Some illustrative examples with linear sorption and decay are presented to demonstrate the application and check the accuracy of the derived analytical solutions. The derived hybrid analytical-numerical solutions are checked against a numerical approach and demonstratively applied to a nonlinear transport example, which simulates a simplified system of iron oxide bioreduction with nonlinear sorption and nonlinear reaction kinetics.
Application of Exactly Linearized Error Transport Equations to AIAA CFD Prediction Workshops
Derlaga, Joseph M.; Park, Michael A.; Rallabhandi, Sriram
2017-01-01
The computational fluid dynamics (CFD) prediction workshops sponsored by the AIAA have created invaluable opportunities in which to discuss the predictive capabilities of CFD in areas in which it has struggled, e.g., cruise drag, high-lift, and sonic boom pre diction. While there are many factors that contribute to disagreement between simulated and experimental results, such as modeling or discretization error, quantifying the errors contained in a simulation is important for those who make decisions based on the computational results. The linearized error transport equations (ETE) combined with a truncation error estimate is a method to quantify one source of errors. The ETE are implemented with a complex-step method to provide an exact linearization with minimal source code modifications to CFD and multidisciplinary analysis methods. The equivalency of adjoint and linearized ETE functional error correction is demonstrated. Uniformly refined grids from a series of AIAA prediction workshops demonstrate the utility of ETE for multidisciplinary analysis with a connection between estimated discretization error and (resolved or under-resolved) flow features.
Energy Technology Data Exchange (ETDEWEB)
Sari, Salih [Hacettepe University, Department of Nuclear Engineering, Beytepe, 06800 Ankara (Turkey); Erguen, Sule [Hacettepe University, Department of Nuclear Engineering, Beytepe, 06800 Ankara (Turkey)], E-mail: se@nuke.hacettepe.edu.tr; Barik, Muhammet; Kocar, Cemil; Soekmen, Cemal Niyazi [Hacettepe University, Department of Nuclear Engineering, Beytepe, 06800 Ankara (Turkey)
2009-03-15
In this study, isothermal turbulent bubbly flow is mechanistically modeled. For the modeling, Fluent version 6.3.26 is used as the computational fluid dynamics solver. First, the mechanistic models that simulate the interphase momentum transfer between the gas (bubbles) and liquid (continuous) phases are investigated, and proper models for the known flow conditions are selected. Second, an interfacial area transport equation (IATE) solution is added to Fluent's solution scheme in order to model the interphase momentum transfer mechanisms. In addition to solving IATE, bubble number density (BND) approach is also added to Fluent and this approach is also used in the simulations. Different source/sink models derived for the IATE and BND models are also investigated. The simulations of experiments based on the available data in literature are performed by using IATE and BND models in two and three-dimensions. The results show that the simulations performed by using IATE and BND models agree with each other and with the experimental data. The simulations performed in three-dimensions give better agreement with the experimental data.
Optimum plane selection for transport-of-intensity-equation-based solvers.
Martinez-Carranza, J; Falaggis, K; Kozacki, T
2014-10-20
Deterministic single beam phase retrieval techniques based on the transport of intensity equation (TIE) use the axial intensity derivative obtained from a series of intensities recorded along the propagation axis as an input to the TIE-based solver. The common belief is that, when reducing the error present in the axial intensity derivative, there will be minimal error in the retrieved phase. Thus, reported optimization schemes of measurement condition focuses on the minimization of error in the axial intensity derivative. As it is shown in this contribution, this assumption is not correct and leads to underestimating the value of plane separation, which increases the phase retrieval errors and sensitivity to noise of the TIE-based measurement system. Therefore, in this paper, a detailed analysis that shows the existence of an optimal separation that minimizes the error in the retrieved phase for a given TIE-based solver is carried out. The developed model is used to derive analytical expressions that provide an optimal plane separation for a given number of planes and level of noise for the case of equidistant plane separation. The obtained results are derived for the widely used Fourier-transform-based TIE solver, but it is shown that they can also be applied to multigrid-based techniques.
Wang, Cong; Long, Yao; Tian, Ming-Feng; He, Xian-Tu; Zhang, Ping
2013-04-01
We have calculated the equations of state, the viscosity and self-diffusion coefficients, and electronic transport coefficients of beryllium in the warm dense regime for densities from 4.0 to 6.0 g/cm(3) and temperatures from 1.0 to 10.0 eV by using quantum molecular dynamics simulations. The principal Hugoniot curve is in agreement with underground nuclear explosive and high-power laser experimental results up to ~20 Mbar. The calculated viscosity and self-diffusion coefficients are compared with the one-component plasma model, using effective charges given by the average-atom model. The Stokes-Einstein relationship, which connects viscosity and self-diffusion coefficients, is found to hold fairly well in the strong coupling regime. The Lorenz number, which is the ratio between thermal and electrical conductivities, is computed via Kubo-Greenwood formula and compared to the well-known Wiedemann-Franz law in the warm dense region.
Directory of Open Access Journals (Sweden)
Kulish Vladimir V.
2004-01-01
Full Text Available This paper presents an integral solution of the generalized one-dimensional equation of energy transport with the convective term.The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of noninteger orders. Confluent hypergeometric functions, known as Whittaker's functions, appear in the course of the solution procedure upon applying the Laplace transform to the original transport equation.The analytical solution of the problem is written in the integral form and provides a relationship between the local values of the transported property (e.g., temperature, mass, momentum, etc. and its flux.The solution is valid everywhere within the domain, including the domain boundary.
Topological Interactions in a Boltzmann-Type Framework
Blanchet, Adrien; Degond, Pierre
2016-04-01
We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However, it exhibits a spatial non-locality instead of the classical non-locality in velocity space. This result relies on the approximation properties of Bernstein polynomials. We illustrate the dynamics with numerical simulations.
Diffusive limit for a quantum linear Boltzmann dynamics
Clark, Jeremy
2010-01-01
We study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model we begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the scattering with the gas particles is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix evolving according to a translation-covariant Lindblad equation. Our main result is a proof that the particle diffuses for large times.
Transport Equations for CAD Modeling of Al(x)Ga(1-x)N/GaN HEMTs
Freeman, Jon C.
2003-01-01
BEMTs formed from Al(x)Ga(1-x)N/GaN heterostructures are being investigated for high RF power and efficiency around the world by many groups, both academic and industrial. In these devices, the 2DEG formation is dominated by both spontaneous and piezoelectric polarization fields, with each component having nearly the same order of magnitude. The piezoelectric portion is induced by the mechanical strain in the structure, and to analyze these devices, one must incorporate the stress/strain relationships, along with the standard semiconductor transport equations. These equations for Wurtzite GaN are not easily found in the open literature, hence this paper summarizes them, along with the constitutive equations for piezoelectric materials. The equations are cast into the format for the Wurtzite crystal class, which is the most common way GaN is grown epitaxially.
Student understanding of the Boltzmann factor
Smith, Trevor I; Thompson, John R
2015-01-01
We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable, nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations on student discussions about the Boltzmann f...
Are there Boltzmann brains in the vacuum
Davenport, Matthew
2010-01-01
"Boltzmann brains" are human brains that arise as thermal or quantum fluctuations and last at least long enough to think a few thoughts. In many scenarios involving universes of infinite size or duration, Boltzmann brains are infinitely more common than human beings who arise in the ordinary way. Thus we should expect to be Boltzmann brains, in contradiction to observation. We discuss here the question of whether Boltzmann brains can arise as quantum fluctuations in the vacuum. Such Boltzmann brains pose an even worse problem than those arising as fluctuations in the thermal state of an exponentially expanding universe. We give several arguments for and against inclusion of vacuum Boltzmann brains in the anthropic reference class, but find neither choice entirely satisfactory.
Quadrature-based Lattice Boltzmann Model for Relativistic Flows
Blaga, Robert
2016-01-01
A quadrature-based finite-difference lattice Boltzmann model is developed that is suitable for simulating relativistic flows of massless particles. We briefly review the relativistc Boltzmann equation and present our model. The quadrature is constructed such that the stress-energy tensor is obtained as a second order moment of the distribution function. The results obtained with our model are presented for a particular instance of the Riemann problem (the Sod shock tube). We show that the model is able to accurately capture the behavior across the whole domain of relaxation times, from the hydrodynamic to the ballistic regime. The property of the model of being extendable to arbitrarily high orders is shown to be paramount for the recovery of the analytical result in the ballistic regime.
Elliptic equation for random walks. Application to transport in microporous media
DEFF Research Database (Denmark)
Shapiro, Alexander
2007-01-01
We consider a process of random walks with arbitrary residence time distribution. We show that in many cases this process may not be described by the classical (Fick) parabolic diffusion equation, but an elliptic equation. An additional term proportional to the second time derivative takes...... into account the distribution of the residence times of molecules ill pores. The new elliptic diffusion equation is strictly derived by the operator approach. A criterion showing where the new equation should be applied instead of the standard diffusion equation is obtained. Boundary conditions are studied...
Li, Qingfeng; Steinheimer, Jan; Petersen, Hannah; Bleicher, Marcus; Stöcker, Horst
2009-04-01
A systematic study of HBT radii of pions, produced in heavy ion collisions in the intermediate energy regime (SPS), from an integrated (3 + 1)d Boltzmann + hydrodynamics approach is presented. The calculations in this hybrid approach, incorporating an hydrodynamic stage into the Ultra-relativistic Quantum Molecular Dynamics transport model, allow for a comparison of different equations of state retaining the same initial conditions and final freeze-out. The results are also compared to the pure cascade transport model calculations in the context of the available data. Furthermore, the effect of different treatments of the hydrodynamic freeze-out procedure on the HBT radii are investigated. It is found that the HBT radii are essentially insensitive to the details of the freeze-out prescription as long as the final hadronic interactions in the cascade are taken into account. The HBT radii RL and RO and the RO /RS ratio are sensitive to the EoS that is employed during the hydrodynamic evolution. We conclude that the increased lifetime in case of a phase transition to a QGP (via a Bag Model equation of state) is not supported by the available data.
Energy Technology Data Exchange (ETDEWEB)
Li Qingfeng [School of Science, Huzhou Teachers College, Huzhou 313000 (China); Frankfurt Institute for Advanced Studies (FIAS), Johann Wolfgang Goethe-Universitaet, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main (Germany)], E-mail: liqf@fias.uni-frankfurt.de; Steinheimer, Jan [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main (Germany)], E-mail: steinheimer@th.physik.uni-frankfurt.de; Petersen, Hannah [Frankfurt Institute for Advanced Studies (FIAS), Johann Wolfgang Goethe-Universitaet, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main (Germany); Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main (Germany)], E-mail: petersen@th.physik.uni-frankfurt.de; Bleicher, Marcus [Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main (Germany)], E-mail: bleicher@th.physik.uni-frankfurt.de; Stoecker, Horst [Frankfurt Institute for Advanced Studies (FIAS), Johann Wolfgang Goethe-Universitaet, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main (Germany); Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main (Germany); Gesellschaft fuer Schwerionenforschung (GSI), Planckstr. 1, D-64291 Darmstadt (Germany)], E-mail: h.stoecker@gsi.de
2009-04-13
A systematic study of HBT radii of pions, produced in heavy ion collisions in the intermediate energy regime (SPS), from an integrated (3+1)d Boltzmann + hydrodynamics approach is presented. The calculations in this hybrid approach, incorporating an hydrodynamic stage into the Ultra-relativistic Quantum Molecular Dynamics transport model, allow for a comparison of different equations of state retaining the same initial conditions and final freeze-out. The results are also compared to the pure cascade transport model calculations in the context of the available data. Furthermore, the effect of different treatments of the hydrodynamic freeze-out procedure on the HBT radii are investigated. It is found that the HBT radii are essentially insensitive to the details of the freeze-out prescription as long as the final hadronic interactions in the cascade are taken into account. The HBT radii R{sub L} and R{sub O} and the R{sub O}/R{sub S} ratio are sensitive to the EoS that is employed during the hydrodynamic evolution. We conclude that the increased lifetime in case of a phase transition to a QGP (via a Bag Model equation of state) is not supported by the available data.
Boltzmann analyses of swarm experiments over the years
Pitchford, Leanne
2013-09-01
Art Phelps was one of the ``grand old men'' in field of gaseous electronics. He was a graduate student when the GEC got started and he attended almost all of the meetings over the years. During his remarkably long career, he produced a number of the classic papers in our field as a glance at Web of Science will show. Art was my mentor and friend, and I had the privilege of working with him for many years on various topics related mainly to electron scattering and transport in weakly ionized gases. In this talk, I will discuss the originality of some of his early work on these subjects in the context of their times, focusing in particular on his publications from the mid-1960's with his colleagues from Westinghouse Research Laboratories. These report the first numerical solutions of the Boltzmann equation for electrons, to my knowledge, and they inspired much subsequent work related to the extraction of quantitative information about low-energy electron scatting with simple gases from measurements of macroscopic parameters (mobility, diffusion,..). I will outline some of the work he and I did together in this topical area using more sophisticated numerical techniques. This and other work in the field eventually led to the establishment of the ongoing GEC Plasma Data Exchange Project which now involves a number of people (the LXCat team), as discussed in Tuesday's workshop. The LXCat team had completed work on noble gases and had just started working on evaluations of cross sections for simple molecules when Art died. We are fortunate to have had his involvement on these projects. Art had ideas for future work in these areas, and some are included in a long e-mail message from Art a couple of years ago that I will share because it includes some suggestions [to the community] for future work.
On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative
Directory of Open Access Journals (Sweden)
Abdon Atangana
2014-01-01
Full Text Available The hydrodynamic dispersion equation was generalized using the concept of variational order derivative. The modified equation was numerically solved via the Crank-Nicholson scheme. The stability and convergence of the scheme in this case were presented. The numerical simulations showed that, the modified equation is more reliable in predicting the movement of pollution in the deformable aquifers, than the constant fractional and integer derivatives.
Momentum Transport in Rarefied Gases.
Hickey, Keith Alan
The study of non-uniform rarefied gas flow under different geometries and boundary conditions is fundamental to problems in a variety of systems. This dissertation investigates problems of viscous flow or momentum transport in the thin regions (Knudsen layers) close to the boundaries where rarefied gas flows must be described by the Boltzmann equation (Kinetic Theory). The problems of planar slip flow and planar Poiseuille flow for rigid spheres are examined by solving the linearized Boltzmann equation using the discrete ordinates (S_{rm N} ) method. The slip flow or half-space problem of rarefied gas flow is considered and use of the S_ {rm N} (discrete ordinates) algorithm outlined. Accurate numerical results for the velocity slip coefficient and velocity defect are obtained for a rigid sphere gas and are compared with previously reported results and experimental data. In plane Poiseuille flow, the continuum limit is characterized by the Burnett distribution. Explicit results for this distribution are obtained by solving numerically the relevant integral equations for a rigid sphere gas in the context of the linearized Boltzmann equation. This distribution together with the Chapman-Enskog distribution is used to obtain asymptotic results (near-continuum) for mass and heat fluxes corresponding to planar thermal transpiration and mechanocaloric effects. The problem of plane Poiseuille flow of a rarefied gas is solved by the S_{rm N } method. Explicit results for the flow rates and velocity profiles for a rigid sphere intermolecular interaction are obtained, and compared with the BGK and one-term synthetic model results. The flow rates are verified by use of variational expressions incorporating the newly developed Burnett distribution values. The rigid sphere values for the flow rates are in better agreement with the available experimental data than those based on the BGK kinetic model and the one term synthetic model. The development of the appropriate equations
Application of the three-dimensional telegraph equation to cosmic-ray transport
Tautz, R C
2016-01-01
An analytical solution to the the three-dimensional telegraph equation is presented. This equation has recently received some attention but so far the treatment has been one-dimensional. By using the structural similarity to the Klein-Gordon equation, the telegraph equation can be solved in closed form. Illustrative examples are used to discuss the qualitative differences to the diffusion solution. The comparison with a numerical test-particle simulation reveals that some features of an intensity profile can be better explained using the telegraph approach.
STUDY ON GAS KINETIC NUMERICAL ALGORITHM USING BOLTZMANN MODEL EQUATION%基于Boltzmann模型方程的气体运动论统一算法研究
Institute of Scientific and Technical Information of China (English)
李志辉; 张涵信
2005-01-01
从Boltzmann-Shakhov模型方程出发,研究确立含流态控制参数可描述不同流域气体流动特征的气体分子速度分布函数方程;研究发展气体运动论离散速度坐标法,借助非定常时间分裂数值计算方法和NND差分格式,结合DSMC方法关于分子运动与碰撞去耦技术,发展直接求解速度分布函数的气体运动论耦合迭代数值格式;研制可用于物理空间各点宏观流动取矩的离散速度数值积分方法,由此提出一套能有效模拟稀薄流到连续流不同流域气体流动问题统一算法.通过对不同Knudsen数下一维激波内流动、二维圆柱、三维球体绕流数值计算表明,计算结果与有关实验数据及其它途径研究结果(如DSMC模拟值、N-S数值解)吻合较好,证实气体运动论统一算法求解各流域气体流动问题的可行性.尝试将统一算法进行HPF并行化程序设计,基于对球体绕流及类"神舟"返回舱外形绕流问题进行HPF初步并行试算,显示出统一算法具有很好的并行可扩展性,可望建立起新型的能有效模拟各流域飞行器绕流HPF并行算法研究方向.通过将气体运动论统一算法推广应用于微槽道流动计算研究,已初步发展起可靠模拟二维短微槽道流动数值算法;通过对Couette流、Poiseuille流、压力驱动的二维短槽道流数值模拟,证实该算法对微槽道气体流动问题具有较强的模拟能力,可望发展起基于Boltzmann模型方程能可靠模拟MEMS微流动问题气体运动论数值计算方法研究途径.
Schneider, Florian
2016-01-01
This paper provides a generalization of the realizability-preserving discontinuous-Galerkin scheme for quadrature-based minimum-entropy models to full-moment models of arbitrary order. It is applied to the class of Kershaw closures, which are able to provide a cheap closure of the moment problem. This results in an efficient algorithm for the underlying linear transport equation. The efficiency of high-order methods is demonstrated using numerical convergence tests and non-smooth benchmark problems.
Tricoli, Ugo; Da Silva, Anabela; Markel, Vadim A
2016-01-01
We derive a reciprocity relation for vector radiative transport equation (vRTE) that describes propagation of polarized light in multiple-scattering media. We then show how this result, together with translational invariance of a plane-parallel sample, can be used to compute efficiently the sensitivity kernel of diffuse optical tomography (DOT) by Monte Carlo simulations. Numerical examples of polarization-selective sensitivity kernels thus computed are given.
Schneider, Florian
2016-10-01
This paper provides a generalization of the realizability-preserving discontinuous-Galerkin scheme given in [3] to general full-moment models that can be closed analytically. It is applied to the class of Kershaw closures, which are able to provide a cheap closure of the moment problem. This results in an efficient algorithm for the underlying linear transport equation. The efficiency of high-order methods is demonstrated using numerical convergence tests and non-smooth benchmark problems.
Lattice-Boltzmann-based two-phase thermal model for simulating phase change
Kamali, M.R.; Gillissen, J.J.J.; Van den Akker, H.E.A.; Sundaresan, S.
2013-01-01
A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A nonideal equation of state (EOS) is introduced by using a pseudopotential LB model. The evolution equation for the pseudotemperature variable is constructed in such a manner that in the continuum l...
Wang, I. T.
A general method for determining the effective transport wind speed, overlineu, in the Gaussian plume equation is discussed. Physical arguments are given for using the generalized overlineu instead of the often adopted release-level wind speed with the plume diffusion equation. Simple analytical expressions for overlineu applicable to low-level point releases and a wide range of atmospheric conditions are developed. A non-linear plume kinematic equation is derived using these expressions. Crosswind-integrated SF 6 concentration data from the 1983 PNL tracer experiment are used to evaluate the proposed analytical procedures along with the usual approach of using the release-level wind speed. Results of the evaluation are briefly discussed.
Sheng, W.; Chen, G.-J.; Lu, H.-C.
1989-01-01
An attempt is made in this work to combine the Enskog theory of transport properties with the simple cubic Peng-Robinson (PR) equation of state. The PR equation of state provides the density dependence of the equilibrium radial distribution function. A slight empirical modification of the Enskog equation is proposed to improve the accuracy of correlation of thermal conductivity and viscosity coefficient for dense gases and liquids. Extensive comparisons with experimental data of pure fluids are made for a wide range of fluid states with temperatures from 90 to 500 K and pressures from 1 to 740 atm. The total average absolute deviations are 2.67% and 2.02% for viscosity and thermal conductivity predictions, respectively. The proposed procedure for predicting viscosity and thermal conductivity is simple and straightforward. It requires only critical parameters and acentric factors for the fluids.
SIMPLE LATTICE BOLTZMANN MODEL FOR TRAFFIC FLOWS
Institute of Scientific and Technical Information of China (English)
Yan Guangwu; Hu Shouxin
2000-01-01
A lattice Boltzmann model with 5-bit lattice for traffic flows is proposed.Using the Chapman-Enskog expansion and multi-scale technique,we obtain the higher-order moments of equilibrium distribution function.A simple traffic light problem is simulated by using the present lattice Boltzmann model,and the result agrees well with analytical solution.
Galanti, Marta; Fanelli, Duccio; Piazza, Francesco
2016-08-01
Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet, often the nature of the constraints coming from many-body interactions or reflecting a complex and confining environment are better understood and modeled at the microscopic level. In this paper we review the subtle link between microscopic exclusion processes and the mean-field equations that ensue from them in the continuum limit. We show that in an inhomogeneous medium, i.e. when jumps are controlled by site-dependent hopping rates, one can obtain three different nonlinear advection-diffusion equations in the continuum limit, suitable for describing transport in the presence of quenched disorder and external fields, depending on the particular rule embodying site inequivalence at the microscopic level. In a situation that might be termed point-like scenario, when particles are treated as point-like objects, the effect of crowding as imposed at the microscopic level manifests in the mean-field equations only if some degree of inhomogeneity is enforced into the model. Conversely, when interacting agents are assigned a finite size, under the more realistic extended crowding framework, exclusion constraints persist in the unbiased macroscopic representation.
Directory of Open Access Journals (Sweden)
Marta Galanti
2016-08-01
Full Text Available Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet, often the nature of the constraints coming from many-body interactions or reflecting a complex and confining environment are better understood and modeled at the microscopic level.In this paper we review the subtle link between microscopic exclusion processes and the mean-field equations that ensue from them in the continuum limit. We show that in an inhomogeneous medium, i.e. when jumps are controlled by site-dependent hopping rates, one can obtain three different nonlinear advection-diffusion equations in the continuum limit, suitable for describing transport in the presence of quenched disorder and external fields, depending on the particular rule embodying site inequivalence at the microscopic level. In a situation that might be termed point-like scenario, when particles are treated as point-like objects, the effect of crowding as imposed at the microscopic level manifests in the mean-field equations only if some degree of inhomogeneity is enforced into the model. Conversely, when interacting agents are assigned a finite size, under the more realistic extended crowding framework, exclusion constraints persist in the unbiased macroscopic representation.
Hydrodynamic Burnett equations for inelastic Maxwell models of granular gases
Khalil, Nagi; Garzó, Vicente; Santos, Andrés
2014-05-01
The hydrodynamic Burnett equations and the associated transport coefficients are exactly evaluated for generalized inelastic Maxwell models. In those models, the one-particle distribution function obeys the inelastic Boltzmann equation, with a velocity-independent collision rate proportional to the γ power of the temperature. The pressure tensor and the heat flux are obtained to second order in the spatial gradients of the hydrodynamic fields with explicit expressions for all the Burnett transport coefficients as functions of γ, the coefficient of normal restitution, and the dimensionality of the system. Some transport coefficients that are related in a simple way in the elastic limit become decoupled in the inelastic case. As a byproduct, existing results in the literature for three-dimensional elastic systems are recovered, and a generalization to any dimension of the system is given. The structure of the present results is used to estimate the Burnett coefficients for inelastic hard spheres.
Implementation of the interfacial area transport equation in trace for boiling two-phase flows
Bernard, Matthew S.
Correctly predicting the interfacial area concentration (a i) is vital to the overall accuracy of the two-fluid model because ai describes the amount of surface area that exists between the two-phases, and is therefore directly related to interfacial mass, momentum and energy transfer. The conventional method for specifying ai in the two-fluid model is through flow regime-based empirical correlations coupled with regime transition criteria. However, a more physically consistent approach to predicting ai is through the interfacial area transport equation (IATE), which can address the deficiencies of the flow regime-based approach. Some previous studies have been performed to demonstrate the feasibility of IATE in developmental versions of the nuclear reactor systems analysis code, TRACE. However, a full TRACE version capable of predicting boiling two-phase flows with the IATE has not been established. Therefore, the current work develops a version of TRACE that is capable of predicting boiling two-phase flows using the IATE. The development is carried out in stages. First, a version of TRACE which employs the two-group IATE for adiabatic, vertical upward, air-water conditions is developed. An in-depth assessment on the existing experimental database is performed to select reliable experimental data for code assessment. Then, the implementation is assessed against the qualified air-water two-phase flow experimental data. Good agreement is observed between the experimental data for ai and the TRACE code with an average error of +/-9% for all conditions. Following the initial development, one-group IATE models for vertical downward and horizontal two-phase flows are implemented and assessed against qualified data. Finally, IATE models capable of predicting subcooled boiling two-phase flows are implemented. An assessment of the models shows that TRACE is capable of generating ai in subcooled boiling two-phase flows with the IATE and that heat transfer effects dominate
Energy Technology Data Exchange (ETDEWEB)
Thompson, Kelly Glen [Texas A & M Univ., College Station, TX (United States)
2000-11-01
In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness
Zhou, Jun; Li, Nianbei; Yang, Ronggui
2015-06-01
The electrons and phonons in metal films after ultra-short pulse laser heating are in highly non-equilibrium states not only between the electrons and the phonons but also within the electrons. An electrohydrodynamics model consisting of the balance equations of electron density, energy density of electrons, and energy density of phonons is derived from the coupled non-equilibrium electron and phonon Boltzmann transport equations to study the nonlinear thermal transport by considering the electron density fluctuation and the transient electric current in metal films, after ultra-short pulse laser heating. The temperature evolution is calculated by the coupled electron and phonon Boltzmann transport equations, the electrohydrodynamics model derived in this work, and the two-temperature model. Different laser pulse durations, film thicknesses, and laser fluences are considered. We find that the two-temperature model overestimates the electron temperature at the front surface of the film and underestimates the damage threshold when the nonlinear thermal transport of electrons is important. The electrohydrodynamics model proposed in this work could be a more accurate prediction tool to study the non-equilibrium electron and phonon transport process than the two-temperature model and it is much easier to be solved than the Boltzmann transport equations.
High-order hydrodynamics via lattice Boltzmann methods.
Colosqui, Carlos E
2010-02-01
In this work, closure of the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) moment hierarchy is accomplished via projection of the distribution function f onto a space H(N) spanned by N-order Hermite polynomials. While successive order approximations retain an increasing number of leading-order moments of f , the presented procedure produces a hierarchy of (single) N-order partial-differential equations providing exact analytical description of the hydrodynamics rendered by ( N-order) lattice Boltzmann-BGK (LBBGK) simulation. Numerical analysis is performed with LBBGK models and direct simulation Monte Carlo for the case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of Weissenberg number Wi=taunuk(2) (i.e., Knudsen number Kn=lambdak=square root Wi); k is the wave number, [corrected] tau is the relaxation time of the system, and lambda approximately tauc(s) is the mean-free path, where c(s) is the speed of sound. The present results elucidate the applicability of LBBGK simulation under general nonequilibrium conditions.
DEFF Research Database (Denmark)
Svec, Oldrich; Skoček, Jan
2013-01-01
The ability of the Lattice Boltzmann method, as the fluid dynamics solver, to properly simulate macroscopic Navier’s slip boundary condition is investigated. An approximate equation relating the Lattice Boltzmann variable slip boundary condition with the macroscopic Navier’s slip boundary conditi...... is proposed. The proposed relation is validated both for the case of Newtonian and non-Newtonian fluids. The importance of employing the Navier’s slip boundary condition is highlighted by a practical industrial problem.......The ability of the Lattice Boltzmann method, as the fluid dynamics solver, to properly simulate macroscopic Navier’s slip boundary condition is investigated. An approximate equation relating the Lattice Boltzmann variable slip boundary condition with the macroscopic Navier’s slip boundary condition...
Yohan, D.; Gerald, D.; Magali, G.; Michel, Q.
2008-12-01
The general problem of transport and reaction in multiphase porous media has been a subject of extensive studies during the last decades. For example, biologically mediated porous media have seen a long history of research from the environmental engineering point of view. Biofilms (aggregate of microorganisms coated in a polymer matrix generated by bacteria) have been particularly examined within the context of bioremediation in the subsurface zone. Five types of models may be used to describe these kinds of physical system: 1) one-equation local mass equilibrium models when the assumption of local mass equilibrium is valid 2) two equations models when the assumption of local mass equilibrium is not valid 3) one equation non-equilibrium models 4) mixed models coupling equations solved at two different scales 5) one equation time-asymptotic models. In this presentation, we use the method of volume averaging with closure to extend the time- asymptotic model at the Darcy scale to the reactive case. Closure problems are solved for simple unit cells, and the macro-scale model is validated against pore-scale simulations.
Filter-matrix lattice Boltzmann model for microchannel gas flows.
Zhuo, Congshan; Zhong, Chengwen
2013-11-01
The lattice Boltzmann method has been shown to be successful for microscale gas flows, and it has attracted significant research interest. In this paper, the recently proposed filter-matrix lattice Boltzmann (FMLB) model is first applied to study the microchannel gas flows, in which a Bosanquet-type effective viscosity is used to capture the flow behaviors in the transition regime. A kinetic boundary condition, the combined bounce-back and specular-reflection scheme with the second-order slip scheme, is also designed for the FMLB model. By analyzing a unidirectional flow, the slip velocity and the discrete effects related to the boundary condition are derived within the FMLB model, and a revised scheme is presented to overcome such effects, which have also been validated through numerical simulations. To gain an accurate simulation in a wide range of Knudsen numbers, covering the slip and the entire transition flow regimes, a set of slip coefficients with an introduced fitting function is adopted in the revised second-order slip boundary condition. The periodic and pressure-driven microchannel flows have been investigated by the present model in this study. The numerical results, including the velocity profile and the mass flow rate, as well as the nonlinear pressure distribution along the channel, agree fairly well with the solutions of the linearized Boltzmann equation, the direct simulation Monte Carlo results, the experimental data, and the previous results of the multiple effective relaxation lattice Boltzmann model. Also, the present results of the velocity profile and the mass flow rate show that the present model with the fitting function can yield improved predictions for the microchannel gas flow with higher Knudsen numbers in the transition flow regime.
Energy Technology Data Exchange (ETDEWEB)
Cartier, J
2006-04-15
This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)
Fractional diffusion equation for heterogeneous medium
Energy Technology Data Exchange (ETDEWEB)
Polo L, M. A.; Espinosa M, E. G.; Espinosa P, G. [Universidad Autonoma Metropolitana, Unidad Iztapalapa, Area de Ingenieria en Recursos Energeticos, Av, San Rafael Atlixco 186, Col. Vicentina, 09340 Mexico D. F. (Mexico); Del Valle G, E., E-mail: plabarrios@hotmail.com [Instituto Politecnico Nacional, Escuela Superior de Fisica y Matematicas, Av. IPN s/n, Col. San Pedro Zacatenco, 07738 Mexico D. F. (Mexico)
2011-11-15
The asymptotic diffusion approximation for the Boltzmann (transport) equation was developed in 1950 decade in order to describe the diffusion of a particle in an isotropic medium, considers that the particles have a diffusion infinite velocity. In this work is developed a new approximation where is considered that the particles have a finite velocity, with this model is possible to describe the behavior in an anomalous medium. According with these ideas the model was obtained from the Fick law, where is considered that the temporal term of the current vector is not negligible. As a result the diffusion equation of fractional order which describes the dispersion of particles in a highly heterogeneous or disturbed medium is obtained, i.e., in a general medium. (Author)
Momentum transport in rarefied gases
Energy Technology Data Exchange (ETDEWEB)
Hickey, K.A.
1989-01-01
The study of non-uniform rarefied gas flow under different geometries and boundary conditions is fundamental to problems in a variety of systems. This dissertation investigates problems of viscous flow or momentum transport in the thin regions (Knudsen layers) close to the boundaries where rarefied gas flows must be described by the Boltzmann equation (Kinetic Theory). The problems of planar slip flow and planar Poiseuille flow for rigid spheres are examined by solving the linearized Boltzmann equation using the discrete ordinates (S{sub N}) method. The slip flow or half-space problem of rarefied gas flow is considered and use of the S{sub N} (discrete ordinates) algorithm outlined. Accurate numerical results for the velocity slip coefficient and velocity defect are obtained for a rigid sphere gas and are compared with previously reported results and experimental data. In plane Poiseuille flow, the continuum limit is characterized by the Burnett distribution. Explicit results for this distribution are obtained by solving numerically the relevant integral equations for a rigid sphere gas in the context of the linearized Boltzmann equation. This distribution together with the Chapman-Enskog distribution is used to obtain asymptotic results (near-continuum) for mass and heat fluxes corresponding to planar thermal transpiration and mechanocaloric effects. The problem of plane Poiseuille flow of a rarefied gas is solved by the S{sub N} method. Explicit results for the flow rates and velocity profiles for a rigid sphere intermolecular interaction are obtained, and compared with the BGK and one-term synthetic model results. The flow rates are verified by use of variational expressions incorporating the newly developed Burnett distribution values. The rigid sphere values for the flow rates are in better agreement with the available experimental data than those based on the BGK kinetic model and the one term synthetic model.
Bohinc, Klemen; Shrestha, Ahis; Brumen, Milan; May, Sylvio
2012-03-01
In the classical mean-field description of the electric double layer, known as the Poisson-Boltzmann model, ions interact exclusively through their Coulomb potential. Ion specificity can arise through solvent-mediated, nonelectrostatic interactions between ions. We employ the Yukawa pair potential to model the presence of nonelectrostatic interactions. The combination of Yukawa and Coulomb potential on the mean-field level leads to the Poisson-Helmholtz-Boltzmann model, which employs two auxiliary potentials: one electrostatic and the other nonelectrostatic. In the present work we apply the Poisson-Helmholtz-Boltzmann model to ionic mixtures, consisting of monovalent cations and anions that exhibit different Yukawa interaction strengths. As a specific example we consider a single charged surface in contact with a symmetric monovalent electrolyte. From the minimization of the mean-field free energy we derive the Poisson-Boltzmann and Helmholtz-Boltzmann equations. These nonlinear equations can be solved analytically in the weak perturbation limit. This together with numerical solutions in the nonlinear regime suggests an intricate interplay between electrostatic and nonelectrostatic interactions. The structure and free energy of the electric double layer depends sensitively on the Yukawa interaction strengths between the different ion types and on the nonelectrostatic interactions of the mobile ions with the surface.
Least-squares finite-element lattice Boltzmann method.
Li, Yusong; LeBoeuf, Eugene J; Basu, P K
2004-06-01
A new numerical model of the lattice Boltzmann method utilizing least-squares finite element in space and Crank-Nicolson method in time is presented. The new method is able to solve problem domains that contain complex or irregular geometric boundaries by using finite-element method's geometric flexibility and numerical stability, while employing efficient and accurate least-squares optimization. For the pure advection equation on a uniform mesh, the proposed method provides for fourth-order accuracy in space and second-order accuracy in time, with unconditional stability in the time domain. Accurate numerical results are presented through two-dimensional incompressible Poiseuille flow and Couette flow.
Static contact angle in lattice Boltzmann models of immiscible fluids.
Latva-Kokko, M; Rothman, Daniel H
2005-10-01
We study numerically the capillary rise between two horizontal plates and in a rectangular tube, using a lattice Boltzmann (LB) method. We derive an equation for the static fluid-solid contact angle as a function of the wetting tendency of the walls and test its validity. We show that the generalized Laplace law with two independent radii of curvature is followed in capillary rise in rectangular tubes. Our method removes the history dependence of the fluid-solid contact angle that had been present in earlier LB schemes.
A lattice Boltzmann method for dilute polymer solutions.
Singh, Shiwani; Subramanian, Ganesh; Ansumali, Santosh
2011-06-13
We present a lattice Boltzmann approach for the simulation of non-Newtonian fluids. The method is illustrated for the specific case of dilute polymer solutions. With the appropriate local equilibrium distribution, phase-space dynamics on a lattice, driven by a Bhatnagar-Gross-Krook (BGK) relaxation term, leads to a solution of the Fokker-Planck equation governing the probability density of polymer configurations. Results for the bulk rheological characteristics for steady and start-up shear flow are presented, and compare favourably with those obtained using Brownian dynamics simulations. The new method is less expensive than stochastic simulation techniques, particularly in the range of small to moderate Weissenberg numbers (Wi).
On space enrichment estimator for nonlinear Poisson-Boltzmann
Randrianarivony, Maharavo
2013-10-01
We consider the mathematical aspect of the nonlinear Poisson-Boltzmann equation which physically governs the ionic interaction between solute and solvent media. The presented a-posteriori estimates can be computed locally in a very efficient manner. The a-posteriori error is based upon hierarchical space enrichment which ensures its efficiency and reliability. A brief survey of the solving of the nonlinear system resulting from the FEM discretization is reported. To corroborate the analysis, we report on a few numerical results for illustrations. We numerically examine some values of the constants encountered in the theoretical study.
Thermal cascaded lattice Boltzmann method
Fei, Linlin
2016-01-01
In this paper, a thermal cascaded lattice Boltzmann method (TCLBM) is developed in combination with the double-distribution-function (DDF) approach. A density distribution function relaxed by the cascaded scheme is employed to solve the flow field, and a total energy distribution function relaxed by the BGK scheme is used to solve temperature field, where two distribution functions are coupled naturally. The forcing terms are incorporated by means of central moments, which is consistent with the previous force scheme [Premnath \\emph{et al.}, Phys. Rev. E \\textbf{80}, 036702 (2009)] but the derivation is more intelligible and the evolution process is simpler. In the method, the viscous heat dissipation and compression work are taken into account, the Prandtl number and specific-heat ratio are adjustable, the external force is considered directly without the Boussinesq assumption, and the low-Mach number compressible flows can also be simulated. The forcing scheme is tested by simulating a steady Taylor-Green f...
An Infinite Restricted Boltzmann Machine.
Côté, Marc-Alexandre; Larochelle, Hugo
2016-07-01
We present a mathematical construction for the restricted Boltzmann machine (RBM) that does not require specifying the number of hidden units. In fact, the hidden layer size is adaptive and can grow during training. This is obtained by first extending the RBM to be sensitive to the ordering of its hidden units. Then, with a carefully chosen definition of the energy function, we show that the limit of infinitely many hidden units is well defined. As with RBM, approximate maximum likelihood training can be performed, resulting in an algorithm that naturally and adaptively adds trained hidden units during learning. We empirically study the behavior of this infinite RBM, showing that its performance is competitive to that of the RBM, while not requiring the tuning of a hidden layer size.
Moving Charged Particles in Lattice Boltzmann-Based Electrokinetics
Kuron, Michael; Schornbaum, Florian; Bauer, Martin; Godenschwager, Christian; Holm, Christian; de Graaf, Joost
2016-01-01
The motion of ionic solutes and charged particles under the influence of an electric field and the ensuing hydrodynamic flow of the underlying solvent is ubiquitous in aqueous colloidal suspensions. The physics of such systems is described by a coupled set of differential equations, along with boundary conditions, collectively referred to as the electrokinetic equations. Capuani et al. [J. Chem. Phys. 121, 973 (2004)] introduced a lattice-based method for solving this system of equations, which builds upon the lattice Boltzmann (LB) algorithm for the simulation of hydrodynamic flow and exploits computational locality. However, thus far, a description of how to incorporate moving boundary conditions, which are needed to simulate moving colloids, into the Capuani scheme has been lacking. In this paper, we detail how to introduce such moving boundaries, based on an analogue to the moving boundary method for the pure LB solver. The key ingredients in our method are mass and charge conservation for the solute spec...
Okamoto, Jun-ichi; Mathey, Ludwig; Härtle, Rainer
2016-12-01
We generalize the hierarchical equations of motion method to study electron transport through a quantum dot or molecule coupled to one-dimensional interacting leads that can be described as Luttinger liquids. Such leads can be realized, for example, by quantum wires or fractional quantum Hall edge states. In comparison to noninteracting metallic leads, Luttinger liquid leads involve many-body correlations and the single-particle tunneling density of states shows a power-law singularity at the chemical potential. Using the generalized hierarchical equations of motion method, we assess the importance of the singularity and the next-to-leading order many-body correlations. To this end, we compare numerically converged results with second- and first-order results of the hybridization expansion that is inherent to our method. As a test case, we study transport through a single-level quantum dot or molecule that can be described by an Anderson impurity model. Cotunneling effects turn out to be most pronounced for attractive interactions in the leads or repulsive ones if an excitonic coupling between the dot and the leads is realized. We also find that an interaction-induced negative differential conductance near the Coulomb blockade thresholds is slightly suppressed as compared to a first-order and/or rate equation result. Moreover, we find that the two-particle (n -particle) correlations enter as a second-order (n -order) effect and are, thus, not very pronounced at the high temperatures and parameters that we consider.
Implementation of the LAX-Wendroff Method in Cobra-TF for Solving Two-Phase Flow Transport Equations
Energy Technology Data Exchange (ETDEWEB)
Salko, Robert K [ORNL; Wang, Dean [ORNL; Ren, Kangyu [University of Massachusetts, Lowell
2016-01-01
COBRA-TF (Coolant Boiling in Rod Arrays Two Fluid), or CTF, is a subchannel code used to conduct the reactor core thermal hydraulic (T/H) solution in both standalone and coupled multi-physics applications. CTF applies the first-order upwind spatial discretization scheme for solving two-phase flow conservation equations. In this work, the second-order Lax-Wendroff (L-W) scheme has been implemented in CTF to solve the two-phase flow transport equations to improve numerical accuracy in both temporal and spatial discretization. To avoid the oscillation issue, a non-linear flux limiter VA (Van Albada) is employed for the convective terms in the transport equations. Assessments have been carried out to evaluate the performance and stability of the implemented second-order L-W scheme. It has been found that the L-W scheme performs better than the upwind scheme for the single-phase and two-phase flow problems in terms of numerical accuracy and computational efficiency.
Reprint of : The Boltzmann--Langevin approach: A simple quantum-mechanical derivation
Nagaev, K. E.
2016-08-01
We present a simple quantum-mechanical derivation of correlation function of Langevin sources in the semiclassical Boltzmann-Langevin equation. The specific case of electron-phonon scattering is considered. It is shown that the assumption of weak scattering leads to the Poisson nature of the scattering fluxes.
Lattice-Boltzmann-based two-phase thermal model for simulating phase change
Kamali, M.R.; Gillissen, J.J.J.; Van den Akker, H.E.A.; Sundaresan, S.
2013-01-01
A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A
Transport solutions of the Lamé equations and shock elastic waves
Alexeyeva, L. A.; Kaishybaeva, G. K.
2016-07-01
The Lamé system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, or supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Numerical results concerning the dynamics of an elastic medium influenced by concentrated transport loads moving at sub-, tran- and supersonic speeds are presented.
Ancey, Christophe; Bohorquez, Patricio; Heyman, Joris
2016-04-01
The advection-diffusion equation arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Stochastic models can also be used to derive this equation, with the significant advantage that they provide information on the statistical properties of particle activity. Stochastic models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. We develop an approach based on birth-death Markov processes, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received little attention. We show that particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particle displacement and the dimensions of the measurement system.